"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Chiarolla, Maria"@en . "2008-12-19T23:59:26Z"@en . "1992"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "The two main questions arising in singular control problems are the characterization\r\nof the boundary of the region of inaction A\u00E2\u0080\u0099 (i.e. the region where it is optimal to\r\ntake no action) and the construction of an optimal control. Among the singular control\r\nproblems the ones in which the class of admissible controls is restricted to the processes\r\nwith monotone non-decreasing components, and the payoff functional does not depend\r\nexplicitly on the control, are usually referred to as monotone follower, cheap control\r\nproblems.\r\nWe identify the free boundary \u00C6\u008CA1 of the two-dimensional monotone follower, cheap\r\ncontrol problem under very mild conditions. We prove that if the region of inaction is\r\nof locally finite perimeter (LFP), then such a region can be replaced by a new region A1 \r\nhaving a more regular boundary. In fact, we show that the new free boundary is\r\ncountably 1-rectifiable and it is also optimal to take no action in the larger set A1. Then\r\nwe give conditions under which the hypothesis (LFP) holds; furthermore we obtain even\r\nhigher regularity of the free boundary, namely C2\u00CE\u00B1, except perhaps at a single corner\r\npoint. This result is easily extended to the n-dimensional case.\r\nUnder the additional hypothesis that the free boundary of the new region of inaction\r\nA1 satisfies a Lipschitz condition (LIP) in a small neighbourhood of the corner point, we\r\nconstruct a control k which acts only when the process is not in A1 and then only to\r\nmove it instantaneously into A1. We show that k is the unique optimal control of the\r\nsingular control problem in question. Finally we give conditions under which (LIP) is\r\nverified. All of these results hold in the n-dimensional case."@en . "https://circle.library.ubc.ca/rest/handle/2429/3236?expand=metadata"@en . "2242340 bytes"@en . "application/pdf"@en . "GEOMETRIC APPROACHTOMONOTONE STOCHASTIC CONTROLbyMaria ChiarollaLaiirea in Matematica, Universit\u00C3\u00A0 degli Studi di Ban, 1982A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYINTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MATHEMATICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril, 1992\u00C2\u00A9 Maria Chiarolla, 1992National Libraryof CanadaBibhoth\u00C3\u00A8que nationaledu CanadaCanadian Theses Service Service des th\u00C3\u00A8es canadiennesOttawa. CanadaKIA 0N4The author has granted an irrevocable non-exclusive licence allowing the National libraryof Canada to reproduce, loan, dsbibute or sellcopies of his/her thesis by any means and inany form or format, maldng this thesis availableto interested persons.The author retains ownership of the copyrightin his/her thesis. Neither the thesis norsubstantial extracts from it may be printed orotherwise reproduced without hislher permission.L\u00E2\u0080\u0099auteur a accord\u00C3\u00A9 une licence irrevocable etnon exclusive pennettant a Ia Bibioth\u00C3\u00A9quenati\u00C3\u00B6nale du Canada de reproduire, pr\u00C3\u00A9ter,distribuer ou vendre des copies de sa thesede quelque manl\u00C3\u00A9re et sous quelque formeque ce soit pour mettre des exemplaires decette these a Ia disposition des personnesint\u00C3\u00A9ress\u00C3\u00A9es.L\u00E2\u0080\u0099auteur conseive (a propci\u00C3\u00A9t\u00C3\u00A9 du droit d\u00E2\u0080\u0099auteurqui protege sa these. Ni (a these ni des extraitssubstantiels de celle-ci ne doivent \u00C3\u00AAtreimprim\u00C3\u00A9s ou autrement reproduits sans sonautorisation.1IiuanaciaISBN 0-315-75398-6In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of I\u00E2\u0080\u0099.A4TLE MAT( C SThe University of British ColumbiaVancouver, CanadaDate AP.(L 2\u00E2\u0080\u0099I-. ifDE-6 (2/88)AbstractThe two main questions arising in singular control problems are the characterizationof the boundary of the region of inaction A\u00E2\u0080\u0099 (i.e. the region where it is optimal totake no action) and the construction of an optimal control. Among the singular controlproblems the ones in which the class of admissible controls is restricted to the processeswith monotone non-decreasing components, and the payoff functional does not dependexplicitly on the control, are usually referred to as monotone follower, cheap controlproblems.We identify the free boundary DA\u00E2\u0080\u0099 of the two-dimensional monotone follower, cheapcontrol problem under very mild conditions. We prove that if the region of inaction isof locally finite perimeter (LFP), then such a region can be replaced by a new regionhaving a more regular boundary. In fact, we show that the new free boundary iscountably 1-rectifiable and it is also optimal to take no action in the larger set A\u00E2\u0080\u0099. Thenwe give conditions under which the hypothesis (LFP) holds; furthermore we obtain evenhigher regularity of the free boundary, namely C2\u00E2\u0080\u009D, except perhaps at a single cornerpoint. This result is easily extended to the n-dimensional case.Under the additional hypothesis that the free boundary of the new region of inactionA\u00E2\u0080\u0099 satisfies a Lipschitz condition (LIP) in a small neighbourhood of the corner point, weconstruct a control k which acts only when the process is not in A\u00E2\u0080\u0099 and then only tomove it instantaneously into A\u00E2\u0080\u0099. We show that k is the unique optimal control of thesingular control problem in question. Finally we give conditions under which (LIP) isverified. All of these results hold in the n-dimensional case.1\u00E2\u0080\u0099Table of ContentsAbstract iiList of Figures vAcknowledgments vi1 Introduction 11.1 Singular Control Problems 11.2 Objective 92 Characterization of the Value Function 112.1 Statement of the Problem 112.2 Some Properties of the Value Function 122.3 A Variational Inequality for the Value Function 273 The Free Boundary 343.1 Introduction 343.2 Identification of the Boundary . . 343.3 A New Approach to Regularity . 483.4 Regularity of the Free Boundary 573.5 Finite Perimeter of A\u00E2\u0080\u0099: a Verification of (LFP) 823.6 Higher Regularity of the Boundary 914 The Optimal Control 1021114.1 Introduction 1024.2 The Optimal Control : a Heuristic Description 1024.3 Construction of The Optimal Control Process 1064.4 Optimality 1324.6 A Verification of (LIP) and (NIN) 137Bibliography 144ivList of Figures3.1 Possible cases (a), (b) and (c) occurring in the proof of Proposition 3.5,taking &2 continuous 433.2 Sketch of the region of inaction A\u00E2\u0080\u0099 and its complement R0 U R1 U R2. . 46VAcknowledgmentsI find my words inadequate to express my gratitude to my supervisor, Prof. Ulrich Haussmann, for his invaluable contribution to my growth as a person and as a researcher. Overthe years Ulrich has been a continual source of inspiration, support and encouragement.I wish to thank him for his friendship, for his vigorous sense of guidance, for the numerous entire afternoons generously spent discussing mathematics, for his patience andunderstanding, and especially for being there when I felt helpless and lost.I wish to thank Prof. Larry Roberts for being supportive at a time when my lifetook a particularly difficult turn. My officemate Yunsun Nam endured me for threeyears and maintained me with her friendship. Ian Lisle efficiently typed Chapter 3(and much more!), but my most profound gratitude goes to Ian for his constant, tactfulcaring and for his many attentions. Chapter 3 was inspired by the work of Prof. RenatoCaccioppoli, who initiated the theory of sets of finite perimeter in 1952. Chapters 1, 2and 4 were courageously typed by Bernardo Hernandez-Morales; his tireless patience inmaking the continuous changes and corrections is gratefully acknowledged. Prof. KeeLam and Ms. Joan de Niverville always took personal interest in matters concerningthe graduate students; their cheerful and competent help is very much appreciated. Mywarmest thanks to Lily Crawford for allowing me to reshape the trees in her backyard ina Mediterranean fashion, and so release my tension.Above all my effort was continuously sustained by my family in Italy. This thesis isdedicated to my parents, Lisetta and Giovannino, with respect and gratitude.This thesis was typeset using LATEX.viChapter 1Introduction1.1 Singular Control ProblemsControl problems for diffusion processes with additive controllable input, when the payofffunctional does not depend explicitly on the control (i.e. the so-called cheap controlproblems) often lead to problems in which the displacement of the state caused by theoptimal control is singular with respect to Lebesgue measure as a function of time. Theoptimal control has usually to be found among controls of bounded variation on finitetime intervals, or, in particular, in the subset of controls whose components are increasingin the time variable. Thus, absolutely continuous controls are allowed as well as impulsecontrols. However, the value function will not be attained at any of these. The optimalcontrol k is singular, and may be characterized by two regions : the region of inactionA\u00E2\u0080\u0099, which is an open set, and its complement (A\u00E2\u0080\u0099)\u00E2\u0080\u0099, the region of action. If the processis in (A, then the control k will make the process jump to some point on the boundaryOA\u00E2\u0080\u0099, thereafter k acts only when the process is on \u00C3\u009CA\u00E2\u0080\u0099 and pushes it back into theregion of inaction A1. The optimal control forces the process to stay inside A\u00E2\u0080\u0099 with aninstantaneous action at the boundary OA\u00E2\u0080\u0099. Therefore k has the peculiarity of the \u00E2\u0080\u009Clocaltime\u00E2\u0080\u009D of the process at the boundary OA\u00E2\u0080\u0099.It should be noticed that such behaviour of the optimal control has really been shownonly for 1-dimensional problems, except for a few very special classes of 2-dimensionalproblems. The reason for this is that several conceptual and technical difficulties arise in1Chapter 1. Introduction 2more than one dimension as we will explain below.Singular control can be approached as the limit of impulse control. Menaldi andRofman [MRfJ study a cheap control problem for n-dimensional diffusion processes whereonly impulse controls are allowed. They obtain the optimal cost as a limit of impulsecontrol problems having a cost for each impulse. The existence of an optimal control isproved only after having restricted the problem to a very particular subset of impulsecontrols. This difficulty suggests already that an optimal control (if there is one) has tobe sought in a much larger set of admissible controls. In fact, Menaldi and Robin [MRbIprove that the infimum of the payoff functional E{f f(Xt)et dt} is continuous as afunction of the initial state, and is the same over the sets of continuous controls, impulsecontrols, or purely jumps controls as long as they are all of locally bounded variation.Menaldi and Robin allow the cost rate f to have polynomial growth, and the diffusionX to be n-dimensional. Moreover, in the 1-dimensional case, they prove the existence ofa singular optimal control k in the class of non-decreasing, non-negative controls, for anon-degenerate diffusion. The optimal process is characterized as the diffusion reflectedat the boundary A1. Moreover, the optimal cost is shown to be C2 and is obtained asthe maximum solution of(All\u00E2\u0080\u0094f)ll = 0 a.e. iniR;e W1\u00E2\u0080\u0099\u00C2\u00B0(1R); All fii.0;(here A is the characteristic operator of the process), where the constraint ll, 0 isformally derived in the limit of impulse control problems when the cost of impulse tendsto zero. A generalization of this result to the 2-dimensional case leads to a variationalChapter 1. Introduction 3inequality of the type(Ai\u00E2\u0080\u0094 f)\u00E2\u0080\u0099\u00C3\u00B1,n2 = 0 a.e. inR2;i=1,2.Again f is supposed to be convex and the process is assumed to have constant drift anddiffusion coefficients (g and o respectively). Then ft is convex, but a new difficulty arises:identifying the region of inaction and proving that it has some regularity.Menaldi and Taksar [MT} study a multi-dimensional problem under the same hypotheses on f, g, u as above, but allow a cost for the control, which enters the equationof the motion additively and is of bounded variation. They approach the optimal costft as a limit of problems with absolutely continuous controls by means of penalization;so they prove that ft is a C\u00E2\u0080\u0099 solution of the Hamilton-Jacobi-Bellman equation of theproblem. The existence and uniqueness of the optimal policy k is also established, butnot in a constructive way. Menaldi and Taksar conjecture, as the reader might havealready guessed, that k should be singular and the optimal process should reflect at OA\u00E2\u0080\u0099in the direction of \u00E2\u0080\u0094Vu.It is now clear that in more than one dimension we have to face problems like thesmoothness of the boundary dA\u00E2\u0080\u0099, and the lack of knowledge of the direction of thereflection. Moreover, even if intuitively we may guess \u00E2\u0080\u0094Vft to be the optimal directionof reflection (since this is the direction of least increase of the cost ft), for some pointson the boundary the gradient Vft may be zero, and so leave the direction of the pushingindeterminate; besides, \u00E2\u0080\u0094Vft need not be among the admissible directions! These areessentially the main problems that have prevented researchers from constructing theoptimal directions and showing the singular feature of the optimal control. The main aimof this work is to overcome this limitation, cf. the statement of objective in Section 1.2.The question of the smoothness of the free boundary 0A1 is studied in a recent paperChapter 1. Introduction 4of Williams, Chow and Menaldi [WCMJ. The state is n-dimensional as in [MT], but thecontrols are monotone non-decreasing and non-negative (component-wise), f is of classC3, and a is non-degenerate. Under the extra hypothesis that and Vf1 never vanishsimultaneously, Williams et al. differentiate the Bellman equation and obtain obstacleproblems solved by the components of the gradient of the value function of theoriginal problem. To these new problems they intend to apply a result of Caffarelli [Cf]concerning the regularity of the free boundary arising in obstacle problems. Caffarelli\u00E2\u0080\u0099sresult guarantees C\u00E2\u0080\u0099-regularity in a neighbourhood of each point p e DA\u00E2\u0080\u0099 of positiveLebesgue density for the coincidence set {i = 0}. However, except for a very special 2-dimensional case (cf. (3.170)), Williams et al. are unable to prove that the points of OA\u00E2\u0080\u0099are points of positive Lebesgue density for the coincidence set. We solve this problem;we show that in their setting, all points of DA\u00E2\u0080\u0099 are of positive Lebesgue density; in fact,the boundary is Lipschitz, except perhaps at one single point (cf. Section 3.6).In some cases, the region of inaction A\u00E2\u0080\u0099 will be a wedge; this is the case for the2-dimensional portfolio model studied by Davis and Norman [DN]. Essential for thedetermination of A\u00E2\u0080\u0099 is the choice of the utility function f. In fact, they choose f(x)equal to x7/\u00E2\u0080\u0099-y or to log x, and from this they deduce the \u00E2\u0080\u009Chomothetic property\u00E2\u0080\u009D and theconcavity of the optimal profit (note that in their paper i is a sup not an inf). The\u00E2\u0080\u009Chomothetic property\u00E2\u0080\u009D allows them to find the form of ii, \u00C3\u00B1 c C2, and prove that A\u00E2\u0080\u0099 isa wedge in the positive orthant of 112. The unique optimal control k turns out to beequal to c,L + c2U where c1, c2 are constant directions, and the 1-dimensional processesL and U are the local times at the lower and upper boundaries respectively. Because ofthe particular shape of the region of inaction A\u00E2\u0080\u0099, the results of Varadhan and Williams[VW] or of Tanaka [Tkj apply and provide the optimal process at least up to the firsthitting time of the corner, which is not finite under the optimal policy.Chapter 1. Introduction 5Sometimes 1-dimensional singular control problems can be reduced to equivalent questions of optimal stopping. In fact, it can be shown that the space derivative of theoptimal cost ft coincides with the optimal risk th of an appropriate stopping problem,whose optimal continuation region is precisely the region of inaction of the control problem. Therefore, by solving the free boundary problem that characterizes the optimalrisk zij, one can obtain bounds on the continuation set of the stopping problem, and soon the inaction region of the singular control problem. This connection between singular stochastic control and optimal stopping was developed rigorously by Karatzas [Krl]mostly by analytical methods based on properties of solutions to free boundary problemsand variational inequalities. Subsequently, Karatzas and Shreve [KS 1] established theconnection between the two problems by using only direct probabilistic arguments. Inparticular, they proved the existence of an optimal control Ic for the control problemwith zero drift, in the setting of the monotone follower problem (i.e., over the set ofnon-decreasing controls), and they showed that the optimal stopping time & for the associated stopping problem is exactly the first time k acts (i.e., & = inf{t : > O}.) Theexistence of & is a straightforward consequence of the equivalence between control andstopping problems, and is obtained relatively easily from the existence of Ic. It shouldbe noticed that control processes are more easily topologized than stopping times, andtherefore Ic may sometimes be obtained by using compactness arguments, while a directproof of the existence of \u00C3\u00B4 may require a number of more or less strong conditions (e.g.Friedman [Fri], van Moerbeke [vM]).The equivalence between control and stopping problems is used by Chow, Menaldi andRobin [CMR] to determine the free boundary 0A1 that separates the regions of action andinaction for a non-stationary control problem, whose 1-dimensional state is governed bya linear S.D.E. possibly degenerate, with time-dependent coefficients, and with controlsto be chosen among non-negative, non-decreasing processes. Chow et al. approach theChapter 1. Introduction 6control problem by a sequence of absolutely continuous control problems; this enablesthem to prove that the optimal cost \u00E2\u0080\u0098\u00C3\u00BC is the unique solution of a certain variationalinequality. Then they construct the optimal control, which is, in fact, Markovian andwhose input produces a reflected diffusion process as the optimal process. To constructsuch reflected diffusion, they assume some regularity of the free boundary.We point out that Chow, Menaldi and Robin also approach the control problem bya sequence of impulse control problems, after having shown that the infimum over theset of impulse controls and the one over the set of Lipschitz controls are both equalto the optimal cost il. However, under the hypothesis of convexity of the cost rate f,the value function 11e of each approximating Lipschitz control problem turns out to beconvex, while the value function ii of each approximating impulse control problem isonly e-convex (see Remark 2.7 for the definition). The convexity of j1e is then used todetermine a variational inequality for the optimal cost \u00C3\u00B1. This is one of the reasons whyseveral authors prefer to approximate by absolutely continuous control problems insteadof impulse control problems. The reader may consult Baldursson [Bd] for conditionsunder which an approximation by absolutely continuous controls is possible, as well asHeinricher and Mizel [HM] for a counterexample. In Heinricher and Mizel\u00E2\u0080\u0099s model thevalue function obtained by minimizing over the set of bounded variation controls isstrictly smaller than the value function corresponding to absolutely continuous controls.The connection with optimal stopping often provides some information about the freeboundary OA\u00E2\u0080\u0099 for the singular control problem in the non-stationary case, as explainedabove, but only for 1-dimensional problems. In fact, in the multi-dimensional case suchconnection is no longer available since the gradient of the optimal cost i is a vector-valuedfunction; therefore, it cannot play the role of the optimal risk for a stopping problem.One might think of taking some directional derivative, i.e. a scalar valued function, butthe choice of a direction will very likely be related to the optimal control. Therefore, itChapter 1. Introduction 7remains an open question how to formulate an opportune stopping problem having someimplications for the control problem.A method somewhat similar to the one described above, in which space-derivativesare taken, is introduced by Soner and Shreve [SS] for a 2-dimensional problem, withconvex cost rate f, zero drift and constant diffusion coefficient, over the set of controls ofthe form j N8d8, where N3 is a unit vector and is a 1-dimensional, non-decreasingprocess. Their method uses the gradient flow of \u00E2\u0080\u0098I to change to a more convenientpair of coordinates, and to obtain a more standard free boundary problem. By usingthis ingenious device, Soner and Shreve characterize the optimal cost l as the uniqueC2\u00E2\u0080\u0094solution of the Hamilton-Jacobi-Bellman (HJB) equation. They also show the freeboundary OA\u00E2\u0080\u0099 to be of class C2\u00E2\u0080\u009D for any c E (0, 1); such smoothness of DA\u00E2\u0080\u0099 is essentialin their construction of the optimal process, which is a 2-dimensional Brownian motionreflected along A\u00E2\u0080\u0099 in the \u00E2\u0080\u0094Vi direction, and is obtained as the unique solution ofthe Skorokhod problem under the necessary conditions of Lions and Sznitman [LS]. Theproof of Soner and Shreve\u00E2\u0080\u0099s result makes critical use of the 2-dimensional nature of theproblem; therefore, it cannot be extended to higher dimensions.So far we have described only two 2-dimensional problems (i.e. [DN], [SS]), and theyboth admit a value function ii of class C2. Let us mention another class of 2-dimensionalproblems for which the optimal cost \u00E2\u0080\u0098IL is of class C2, and a connection with optimalstopping is still available. This is the class of finite-fuel problems; these are obtainedby studying a 1-dimensional Brownian motion under a constraint on the total variationof the control process, so that a reduction to a 2-dimensional problem is possible if onedefines the second state variable y to be the remaining fuel. Therefore, one ends updealing with a 2-dimensional problem in which, however, the second state variable isnot a diffusion, but stays constant until the time the control acts, and at that time itdecreases an amount equal to the displacement caused by the control. That is indeed aChapter 1. Introduction 8very special class of 2-dimensional problems.The finite fuel monotone follower problem is solved analytically by Chow, Menaldiand Robin [CMR] and probabilistically by Karatzas [Kr2]. The optimal policy is shownto behave as if one possessed an infinite amount of fuel, until the supply is exhausted,then no further control is exercised. This is justified by the fact that in the monotonefollower setting, fuel causes displacement only in one direction, and the displacementcaused by a unit of fuel is the same whenever it is used; hence there is no point in savingfuel. In particular, Karatzas proves that the directional derivative of the optimal costn(x, y) in the 45\u00C2\u00B0-direction is the optimal risk zI of a suitable stopping problem, andif the control k(y) is optimal for the finite-fuel problem with available fuel y, then therandom times &(y) := inf{t > 0 : (y) > 0}, y > 0, are all optimal in the stoppingproblem. The specific selection of the 45\u00C2\u00B0-direction can be understood by observing thata control k pushes the 2-dimensional state process (Xe, Y) in the (\u00E2\u0080\u00941, \u00E2\u0080\u00941) direction, sinceX = x + W \u00E2\u0080\u0094 k, Y = y \u00E2\u0080\u0094 kt; therefore the directional derivative of il in the oppositedirection may be associated with a stopping problem, by analogy with the 1-dimensionalcase.We conclude by recalling a different type of finite-fuel monotone follower problem, inthe class of cheap control problems with a finite horizon. This is considered by Sun andMenaldi [SM]; their model is characterized by a non-linear, degenerate, 2-dimensionalsystem of the formf dX = YdY = g(X,Y)dt + ydW +dkwhere g is non-linear, \u00E2\u0080\u0098y > 0, and the increasing control k is limited by a finite sourceof fuel (so none of the variables measures the remaining fuel, in contrast to the previousmodels). The optimal cost il is shown to be the maximum solution of a variationalinequality, in the a.e. sense and in the viscosity sense (see Lions [Lnj for the definitionChapter 1. Introduction 9of viscosity solution), and a solution of a sort of Hamilton-Jacobi-Bellman equation; theuniqueness is still an open problem. For the special case of a linear system (i.e. glinear) and a convex cost rate, and under the assumption that an optimal control for thelimit-free problem (i.e. the problem with unlimited fuel supply) exists, Sun and Menaldi[SM] obtain the same kind of optimal policy for the finite-fuel as we have seen for thestandard 1-dimensional finite-fuel monotone follower problem: behave as if you possessedan infinite amount of fuel until you run out of it, then use no further control.1.2 ObjectiveThe objective of this thesis is to derive, in a constructive way, the existence of the optimalcontrol for a 2-dimensional, monotone follower, cheap control problem. Our assumptionson the cost rate f are similar to the ones in [CMR], [MT].In Chapter 2 we state the control problem and we recall the main properties of thevalue function i. In Chapter 3 we identify the free boundary \u00C3\u009C\u00C3\u0084\u00E2\u0080\u0099 of the region of inactionand the three regions R0, R,, R2 into which the region of action splits. Then we assumethat the region of inaction A\u00E2\u0080\u0099 is of locally finite perimeter (i.e. (LFP)) and we showthat A\u00E2\u0080\u0099 can be replaced by a new region of inaction A\u00E2\u0080\u0099, without affecting the originalcontrol problem. We prove that the new free boundary DA\u00E2\u0080\u0099 is countably 1-rectifiable.All of these properties are deduced from the geometry of the problem. To justify theassumption of (LFP) we show that (LFP) is verified under the conditions of [WCM];moreover, in this case, we prove that the free boundary is of class C2\u00E2\u0080\u0099 and the valuefunction is of class C3 up to the boundary (except perhaps at one single point P0 ofIn Chapter 4 we construct a \u00E2\u0080\u009Cpotentially\u00E2\u0080\u009D optimal control k by making use of thegeometric properties of deduced in the previous chapter. Then, under the hypothesisChapter 1. Introduction 10that in a small neighbourhood of P0 the boundary satisfies a Lipschitz condition, we provethat k is the unique optimal control of the singular problem in question. We show thatk acts like the \u00E2\u0080\u009Clocal time\u00E2\u0080\u009D of the optimal process at the boundary OA\u00E2\u0080\u0099; k has constantdirection on each of the two branches O, 02 of 0A1, these directions being parallel to thesides of the wedge R0={i = 0, i = 1, 2}. Finally we give examples in which, near Fo,the required Lipschitz condition is satisfied, and hence k is optimal.Chapter 2Characterization of the Value Function2.1 Statement of the ProblemLet (, F, F) be a probability space, (We, t 0) a standard 2-dimensional Brownianmotion, and (Fe, t 0) a filtration satisfying the usual conditions with respect to W(i.e. F is an increasing, right-continuous family of completed u-subalgebras of F, andW is a martingale with respect to J).We now introduce some notation that will beused throughout this thesis.\u00E2\u0080\u00A2 x+ :=(xt,4) if x = (x1,2)e 112, with xt := max{0,x2}.\u00E2\u0080\u00A2 g is a constant vector in 112; o is a constant 2 x 2 matrix (possibly degenerate).\u00E2\u0080\u00A2 := {(x1,x2)E 112 : x1 0,x2 o}.\u00E2\u0080\u00A2 A function k with values in 112 is A*nondecreasing if k2\u00E2\u0080\u0094 k1 E A* whenevertl1.\u00E2\u0080\u00A2 t(x) := (\ + (x(2)-P-2, 11(x) := ( + IxI2)\u00E2\u0080\u0099, x E 112, with 0 < < 1 and p> 1.11Chapter 2. Characterization of the Value Function 12\u00E2\u0080\u00A2 L is the set of functions v on JR2 such that ,uv2 is integrable.\u00E2\u0080\u00A2 W:={v:vEL, IVvIeL}.\u00E2\u0080\u00A2 G is the set of locally Lipschitz continuous functions v on JR2 such that v(x)IC(1 + xl)\u00E2\u0080\u009D, and IVv(x)l C(1 + IxlY\u00E2\u0080\u0099 a.e. x, for some constant C, with p> 1.The state of the system is described by the following Ito equation:(dX = gdt+ u\u00E2\u0080\u00A2 dW + dk(2.1)(X = x + k0where x E JR2 is the initial state of the control-free diffusion. Note that sometimes wewill write X(t) instead of X when we want to emphasize the dependence of X on theinitial point x. The cost associated with each initial position x E ]R2 and each controlk V is given by(2.2) J(k) = E {j\u00C2\u00B0\u00C2\u00B0 f(X)etdt}where p> 0 is a discount factor, and the cost rate f is a strictly convex, non-negativefunction such thatf(x)\u00E2\u0080\u0094f +oo as lxi \u00E2\u0080\u0094 +00.lxiThe value function is(2.3) \u00E2\u0080\u0098i%(x) = inf{J(k) : k v+}.Our goal is to construct a process k in V for which \u00C3\u00B1(x) = J(k).2.2 Some Properties of the Value FunctionHere we recall some basic properties of the optimal cost \u00C3\u00B1 obtained already in [CMR]for the 1-dimensional case, and in [MT] for the n-dimensional case. For the cost rate fwe assume coercivity (see (2.4) below) besides hypotheses as in the mentioned papers,Chapter 2. Characterization of the Value Function 13i.e. we assume that there exist p> 1 and constants 0 < r Co, C1,C2 such that for anyA E (0,1), any x E 112 and any x\u00E2\u0080\u0099 such that Ix\u00E2\u0080\u0099I 1,(2.4) rxj\u00E2\u0080\u0099\u00E2\u0080\u0094 Co f(x) Co (1+ IxI),(2.5) f(x) - f(x + x\u00E2\u0080\u0099)I c1(i + f(x) + f(x +(2.6) 0 < f(x + Ax\u00E2\u0080\u0099) + f(x \u00E2\u0080\u0094 Ax\u00E2\u0080\u0099) \u00E2\u0080\u0094 2f(x) C2A(1 + f(x)),with q (1\u00E2\u0080\u00942/p)+. (The first part of (2.4) is needed only in the case of cheap controlproblems). An example is f(x) = IxIP with p E IN, p even.Theorem 2.1 There exist constants C0, G1, G2, such that for each A E (0,1) and eachx\u00E2\u0080\u0099 with x\u00E2\u0080\u0099I < 1 the function ii(x) satisfies (2.7)\u00E2\u0080\u0094(2.9) below(2.7) 9xj\u00E2\u0080\u009D\u00E2\u0080\u0094 Oo (x) \u00C3\u00A30(1 +IxI)\u00E2\u0080\u009D,(2.8) Ii(x) \u00E2\u0080\u0094 (x + x\u00E2\u0080\u0099)I O1(1 + xI + x + x\u00E2\u0080\u0099I)\u00E2\u0080\u0099Ix\u00E2\u0080\u0099I,(2.9) 0 i(x + Ax\u00E2\u0080\u0099) + (x \u00E2\u0080\u0094 Ax\u00E2\u0080\u0099) \u00E2\u0080\u0094 2i(x) < 2A(1 + IxI)2.Proof: The proof is as in [MT]. Recall that i(x) inf{J(k) : k E V}, so in particularfor k 0 we get from (2.4)n(x) J(0) = E{f f(x + gt + u Wt)etdt}E{jCo(1 +x+gt+u. WtJP)e_Ptdt}+CoE{fIx+gt+u.WtI1)e_)tdt};but g and u are constant, so we have(2.9)\u00E2\u0080\u0099 E{j Ix + yt + a e1tdt} 0 then weapply Holder\u00E2\u0080\u0099s inequality to get12(x + Ax\u00E2\u0080\u0099) + i(x \u00E2\u0080\u0094 Ax\u00E2\u0080\u0099) \u00E2\u0080\u0094 2ft(x)2/pC2A sup{ (f etdt) (E {f (1 + f(X(t)))et}) }C2Asup{(+ f f(X(t))eutdt)l2?h1}1 1-\u00E2\u0080\u0098l\u00E2\u0080\u00942/pC2A\u00E2\u0080\u0094--(_ + Co(1 + IxDjsince we may restrict ourselves only to controls k satisfying (2.10). Then, from the lastinequality above the second part of (2.9) follows. \u00E2\u0080\u00A2Since ii is convex, i exists a.e. in ]R2. From the growth conditions (2.7)\u00E2\u0080\u0094(2.9) wecan actually obtain more regularity for Il. This is done by an application of Sobolev\u00E2\u0080\u0099sinequality.Chapter 2. Characterization of the Value Function 17Proposition 2.2 The optimal cost i% is inProof: From (2.7)\u00E2\u0080\u0094(2.9) we deduceo n(x) O0(1 + xI)P,(2.11) I(x)I O(1 + IxI)P1, a.e.,o D21t(x)x\u00E2\u0080\u0099 x\u00E2\u0080\u0099 C2(1 + IxI)(2 a.e., for Ix\u00E2\u0080\u0099I 1,(here Oi is slightly greater than the one in (2.8), as necessary.)We remark that the matrix D2(x) exists a.e. and is symmetric. In fact, since %is convex, the surface (x, n(x)) satisfies Euler\u00E2\u0080\u0099s Theorem a.e., and at any point (q, \u00C3\u00B1(q))where Euler\u00E2\u0080\u0099s Theorem holds, one hasi(x) = (q) + (x- q) . V(q) + B(q)(x- q) (x- q) + o(Ix- q12)where B(q) is symmetric and equals D2\u00C3\u00BC(q) (cf. [Bs]).From (2.11)3 with x\u00E2\u0080\u0099= (e1,2) we have0 n11(x) +2i12(x) + i2(x)e C2(1 + IxI)(2)+, a.e..In particular, for j = 1, 2 = 0, and = 0, 2 = 1 we have0 \u00C3\u00B1(x) C2(1 + IxI)(2)+, a.e..Also, for i = = 1, we get12(x) + 22(x) \u00E2\u0080\u0094 C2(1 + x)2 2i12(x) 11(x) +\u00C3\u00B122(x), a.e.,that is2In12(x)I 2O(1 + IxI)(2)+, a.e.HenceI2(x)I = Iu21(x) C2(1 + IxI) \u00E2\u0080\u0098 ,2ooand u E W0 follows.Chapter 2. Characterization of the Value Function 18Theorem 2.3 There is a version of which is in C\u00E2\u0080\u0099(R2).Proof: From Proposition 2.2 we also have \u00C3\u00B130 E W2\u00E2\u0080\u0099\u00C2\u00B0\u00C2\u00B0 since I(8)I (p + 1)3 andI(i)I (p + l)(p + 4)/3d. Now by Sobolev\u00E2\u0080\u0099s embedding theoremw2\u00C2\u00B0\u00C2\u00B0\u00C3\u00A7iR) \u00E2\u0080\u0098\u00E2\u0080\u0094. C\u00E2\u0080\u009D(R2),and therefore there is a version of it/3 in C\u00E2\u0080\u009D(]R2). Thus a version of in C\u00E2\u0080\u0099(1R2) isobtained. \u00E2\u0080\u00A2In order to characterize the optimal cost ii as the solution of a suitable variationalinequality, we introduce two different kinds of perturbed problems approximating Il byabsolutely continuous controls, and by impulse controls respectively. Only the first classof problems is needed to obtain a variational inequality for , but we will describe bothof them for completeness and also because the connection among these problems is interesting in its own right.Let V be the set of impulse controls k in V, i.e.= {k E V stopping times{8j}jEN with 0 O 0+i,j E IN,(2.12) .F9-measurable random variables {j}3 such thatk(t)=for every t >let Ve be the set of Lipschitz continuous controls k in V with Lipschitz constant boundedby 1/ ( > 0), i.e.(2.13) v = {k E V : k is a.s. Lipschitz continuous with Lipschitz constantalso, we set V0= U V. We now define the approximating problems(2.14) = inf{J(k) : k E V},(2.15) x) = illf{J(k) +E[e6i] : k v}.Chapter 2. Characterization of the Value Function 19Theorem 2.4 (x) = inf{J(k) : k e v0} = inf{J(k) : k e V}.Proof: The proof of this theorem is still based on the density of V0 in V as in [CMR],[MTI, or [MRb]; nevertheless, the arguments of the first and second paper do not apply toour problem because of our infinite horizon and cheap monotone control feature (resp.),while the reasoning of the third paper makes essential use of analytical methods. Here wegive a purely probabilistic proof by borrowing some of the ideas of the previous papers.(i) Fix x e 112. Let k\u00E2\u0082\u00ACV be such that J(k) 0,i sitk(O) = k(0) A n \u00E2\u0080\u0094+since k(t) is cadlag. Therefore, k(t) \u00E2\u0080\u0094* k(t) as n \u00E2\u0080\u0094* oc for all t except for the countableset of points where k(t) k(t\u00E2\u0080\u0094) (as guaranteed by the monotone property of k(t)).Moreover, we have a.s.I -k,(t) <2n2(2.17) dt \u00E2\u0080\u0094( k)(t) is non-decreasing in t,since k(t) is non-decreasing. Thus, k E V0 for each n.To simplify the notation we denote by X and X the trajectories starting at x andassociated with the controls k and k (respectively). Let R > 0, then we define thestopping times(TR = inf{t : IX(t)I R},(2.18)11=inf{t:IX(t)IR}, n\u00E2\u0082\u00ACIN,Chapter 2. Characterization of the Value Function 20and we set(2.19) u(R, n) = \u00E2\u0080\u0098rR A TR.Now we modify the controls k as follows(k(t) iftO R\u00E2\u0080\u0094*oo R\u00E2\u0080\u0094ooand therefore inf\u00C3\u00B1e(x) = n(x), since k E V is arbitrary. Let us denote by X\u00E2\u0080\u009D thetrajectory corresponding to k,a; then,J(k,R) = E {J\u00C2\u00B0\u00C2\u00B0 f(flR(t)) etdt}(2.22)f(X(t)) e_Ptdt} + E{J) fR(t))e_Ptdt}Claimi. P{limrR 0 (see the proof of Theorem 2.1),thereforeE{Lfl) f(-:R(t))e_Ptdt}E{Co(l + Ix(u(R,n))I)e(\u00E2\u0080\u0099}Oo(1 + 8R)E {e_P0(Rn)}O0(1 + 8R)P{E {e_PTR a(R, n) = TR} P {a(R, n) = TR}+ E {e\u00E2\u0080\u0099) a(R, n) = T\u00E2\u0080\u0099R} . P {TR> r8R}]O0(i + 8R)T\u00E2\u0080\u0099[E {e_PT} + P {TR> TR}].Clearly, in (2.21) it suffices to consider oniy controls k such that J(k) <+00. Hence byinvoking again the strong Markov property and the dominated convergence theorem, weobtainE {e_PT(Xx(rR))} E {j f(X(t))edt} . 0 as R \u00E2\u0080\u00A2\u00E2\u0080\u0098 00,since TR \u00E2\u0080\u0094* oo a.s.. Therefore it follows from the coercive property (2.7) of i thaturn E {i9X(rR)I1e_1T} = 0. Now we observe that k()(t) 0 (i = 1,2) impliesR\u00E2\u0080\u0094+ooX(t)I IX(t)I + IX(t)\u00E2\u0080\u0094 k(t)I; therefore we will conclude thatE {IR)e_19T1} E { IXz(TR)1\u00E2\u0080\u0099 e_PTR} 0, as R cc,if we show(2.27) Urn E{jX(rR) \u00E2\u0080\u0094 k(Tft)IPe_mR} = 0.R-*ooSince g, u are constant, (2.9)\u00E2\u0080\u0099 holds and so lim E{IX(t) \u00E2\u0080\u0094 k(t)ePt} = 0; hence, byt\u00E2\u0080\u0094oopassing to a subsequence, we have lim E {Ix(t)\u00E2\u0080\u0094 k(t)I\u00E2\u0080\u009D e\u00E2\u0080\u009Dti} = 0. [n order to show3\u00E2\u0080\u0094400(2.27) we setY(t) :=X(t)\u00E2\u0080\u0094k(t)=x+gt+o.W,Chapter 2. Characterization of the Value Function 23and we observe that for each j,urn E{IY(TR A tj)Ie_1\ui} = E{IY(tj)I)e_Iti}since a.s., TR \u00E2\u0080\u0094* co as R \u00E2\u0080\u0094* co and the dominated convergence theorem applies becauseof the estimateE{supIY(t)I} c(p,t,a,g)(1 + IxI)ttiwith c(p, t, a, g) constant. Now we apply the Ito formula to IY(tW)e_Pt from TR A t toTR and we havelim SUPE{IY(TR A \u00E2\u0080\u0094 IY(TR)Ie_PTR}IR>O= lim supE{JR {_c(p,a)IY(t)I2 \u00E2\u0080\u0094 c(p,g)Y(t)P_l +pIY(t)IP]etdt}2\u00C2\u00B0\u00C2\u00B0 R>O TftAi3+ E{JR c_Pt Y(t)I2a dW} ,TRAtiwhere c(p,a) and c(p,g) are constants. But E{f (IY(t)V\u00E2\u0080\u0099ehluI)2dt} < c\u00E2\u0080\u0099o (bymeans of an estimate like (2.9)\u00E2\u0080\u0099), hence the expectation of the martingale above equalszero and solim supE{IY(rR A tj)IPe_TRti \u00E2\u0080\u0094 IY(TR)IPe_PTh1}2\u00C2\u00B0\u00C2\u00B0 R>Olim sup E{f 1[TRAtJ,TR](t)[c1Y(t)IP_2 +c2IY(t)\u00E2\u0080\u0099 +c3IY(t)Ije_Pt dt}3_00R>O 0.lirnE{f 1[tj,)(t)[c1Iy(t)IP2 +c2Y(t)\u00E2\u0080\u0099 +c3IY(t)IP]e_Ptdt}where c1, c2, c3 are positive constants and we have used the fact that TR \u00E2\u0080\u0094* 00 as R \u00E2\u0080\u0094* cc.Now an estimate like (2.9)\u00E2\u0080\u0099 allows us to apply the dominated convergence theorem, andhence (since tj \u00E2\u0080\u0094* cc as j \u00E2\u0080\u0094* cc) we obtainlim E{IY(TR A =3 \u00E2\u0080\u0094+00uniformly in R.Chapter 2. Characterization of the Value Function 2400. We definet< leN, n EN.sup k0() \u00E2\u0080\u0094 k0(t), a.s.; so for any T > 0,lENsup Iv(t) \u00E2\u0080\u0094 k0(t)I < sup sup k0(.L) \u00E2\u0080\u0094 k0(t) \u00E2\u0080\u0094* 0O 0 and R R,(2.28) Co (1 + 8R)\u00C2\u00B0 E {e_PT1} T8R} <;from (2.28), (2.29) we obtainE{J f(X1(t))e_Ptdt} <2E, n > n(R),o(R,n)but is arbitrarily small, so Claim 3 is proved. Now from Claims 1\u00E2\u0080\u00943, (2.21) follows, andso does the first part of the theorem.(ii). Let k0 C V0 such that J(k0) <(2.30) v(t) = k0(--) ifClearly v C V and Iv(t) \u00E2\u0080\u0094 k0(t)I(2.31)Chapter 2. Characterization of the Value Function 25as n \u00E2\u0080\u0094* 00, a.s., since k0 is continuous. Let X and Xr be the trajectories associatedwith the controls i/n and k0, respectively. Let R> 0 and defineTR = inf{t: X(t)I R}, T = inf{t : 1X(t)I R}.As in the proof of (i) we set(2.32) u(R, ii) = TR A r,and we define(v(t) if t 0, then by the definition of infimum and Theorem 2.4 there exists a controlk E V. such thatJ(k) 0;2 Dxthis operator corresponds to a new problem involving a 2 x 4 matrix a3 [a, 61] (witha3 = aa* + 52J non-degenerate) and a 4-dimensional standard Brownian motion W6.If we call \u00C3\u00B1 the optimal cost of the new problem, then we have- 1/th4\\u00E2\u0080\u0094 1/th4\_(2.45) A3u +\u00E2\u0080\u0094\u00C3\u00A7\u00E2\u0080\u0094\u00E2\u0080\u0094) + \u00E2\u0080\u0094iv\u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094) = f, 6> 0;E0x1 E0X2also, satisfies conditions (2.7)\u00E2\u0080\u0094(2.9) (the proof is the same as in Theorem 2.1), andthe estimates hold uniformly in 6, for 6 small. Therefore, 14, V14 and A14 are locallybounded in x, uniformly in 6, and 14 E W(R2), which is reflexive. So there existsa subsequence 6k \u00E2\u0080\u0094 0 such that Iz14(x)I is locally bounded in x, and 14(x) \u00E2\u0080\u0094 v(x),Vi4(x) \u00E2\u0080\u0094* Vv(x) locally uniformly in x, and also A14 \u00E2\u0080\u0094* Av in the sense of distributions.Then, for 6k \u00E2\u0080\u0094f 0 in (2.45) we obtain (2.44) for v. We still have to show that v = . Inorder to do that, we observe that for every control k e V(2.46) E {X(t) \u00E2\u0080\u0094 X(t)} = E {[a, 61] . \u00E2\u0080\u0094 a\u00E2\u0080\u00A2 w} C6\u00E2\u0080\u0099\u00E2\u0080\u00992for some constant C > 0, if X3 and X are the diffusions associated with [a, 61] anda respectively. Now (2.46) and the estimates (2.4), (2.5) imply v = in the limit as* 0.(Uniqueness). The uniqueness of the solution to (2.44) follows from the weak maximumprinciple for degenerate elliptic equations (compare with the proof of Theorem 2 in [MT],p.226).Chapter 2. Characterization of the Value Function 29Assume uE and v are solutions of (2.44). SetIIII = max{i, 2I} for = (1,2) E R2Ao=A-pI;(x,A) = (A+ xI), p> 1,with A to be chosen later and p as in (2.7);W(x) u(x)\u00E2\u0080\u0094 v(x);w(x) = W(x)ib(x, A).Then, since e and v satisfy (2.7) (cf. Remark 2.7), certainly urn w(x) = 0; hence ifIxl-*oowe suppose w(x) 0 and, for example, w(x) > 0 at some point x, then there exists x0such thatw(x0) = maxw(x) > 0.Since w Er12oo(1R) (by the same arguments as in Proposition 2.2), Aw is defined a.e..Let N be the set of zero measure where Aw is not defined; let {zn}neN be a sequencesuch that(Zn_*XO asn\u00E2\u0080\u0094*oo,(2.47)I Zn (N)\u00E2\u0080\u0099 for every n E IN;then we haveAow(z) =A0W(Zn)(Zn, A) + W(Zn)AO(Zn, A) \u00E2\u0080\u0094 (uu*vW(z)) V(z, A)r 1 Ou \u00E2\u0080\u0094 1 due\u00E2\u0080\u0094 1 thI \u00E2\u0080\u0094 1 dye\u00E2\u0080\u00941= [(Zn)) _(_-_()) +_(\u00E2\u0080\u0094(z)) -i-\u00E2\u0080\u0094(\u00E2\u0080\u0094\u00E2\u0080\u0094(Z)) ]\u00E2\u0080\u0098J)(zA)\u00E2\u0080\u0094 pW(Zn)\u00E2\u0080\u0099IJ(Zn, A) + W(Zn)AOV)(Zn, A)+ (au*w(Zfl)-VW(Zn)). V,A).(2.48)Chapter 2. Characterization of the Value Function 30From the dynamic programming equation follows1 \u00C3\u00B6 \u00E2\u0080\u0094 1 \u00C3\u00B6 \u00E2\u0080\u0094 1 0? \u00E2\u0080\u0094 1 OVE \u00E2\u0080\u0094__Q__(z))\u00E2\u0080\u0094 _(__(z)) + + _(\u00E2\u0080\u0094(z))= min{W(zn) - min{Vv(zn)= mm max {Vu(z) \u00E2\u0080\u0094 Vve(z)2 49 IIII1/e II?7II1/emax{VW(zn)1[OW +0W += _[(\u00E2\u0080\u0094_(z)) (-.__(z))= 7(Zn) VW(z)with 7 = (71,72) given byIi if->0,7j(Z)= oxt\u00E2\u0080\u0094\u00E2\u0080\u0098 0 otherwise.Hence (2.48) and (2.49) implyA0w(z). VW(z)(z, A)\u00E2\u0080\u0094 V(z, A)\u00E2\u0080\u0094 pW(Zn)+ W(z)[A0(z,A) + V(z,A)](2.50)= 7(Zn) [_W(z)V(z, A) + Vw(zn)](zn, A)\u00E2\u0080\u0094 pW(Zn)\u00E2\u0080\u0094 gJ*Vw(z) + W(z)6(z, A)(z, A),\u00E2\u0080\u0098(z, )since from(2.51) o asA+oofollowsAo(z,A) + . V(z,A) = 6(z,A)(z,A)b(z, )with(2.52) sup 15(x,A)I \u00E2\u0080\u0094*0 as A \u00E2\u0080\u0094 +oc.Chapter 2. Characterization of the Value Function 31Then we haveAow(z) [_Ly(zn)v(zn,X)\u00E2\u0080\u0094 p+ 6(z,)]w(z)(2.53) +[_aa*Z +since for n sufficiently large w(z) 0.But \u00E2\u0080\u0098y is bounded and = + is bounded as \u00E2\u0080\u0094* +00, uniformlyin n (since W(xo) 0 and W e C\u00E2\u0080\u0099(1R2)); hence we can choose ; > 0 large enough toguarantee(2.54) +6(zn,)_a*\u00C2\u00B03). <\u00E2\u0080\u00A2Therefore, (2.53) and (2.54) implyAow(z) < \u00E2\u0080\u0094w(z),hence(2.55) Urn Aow(z) \u00E2\u0080\u0094w(x0)<0.n\u00E2\u0080\u0094*ooSince (2.55) holds true for any sequence satisfying (2.47), we conclude thatess-lim Aow(x) <0,but this contradicts the weak maximum principle (cf. [Bn], Theorem 1, p.334).We now take \u00E2\u0080\u0094* 0 and obtain a variational inequality for ii.Theorem 2.11 The optimal cost ii is the maximal solution ofAuf a.e. inlR2;2\u00E2\u0080\u00940 a.e. inR;(2.56) Ox1 Ox2(Au\u00E2\u0080\u0094 f)-.- = 0 a.e. in 1R2G02uL\u00C2\u00B0\u00C2\u00B0uE p\u00E2\u0080\u0099 E locChapter 2. Characterization of the Value Function 32Proof: We first observe that we have uniform boundedness of, IVI, andin L, L, and L respectively because (as we pointed out in Remark 2.7) the estimates(2.7)\u00E2\u0080\u0094(2.9) hold for , uniformly in \u00E2\u0082\u00AC, e E (0, 1]. Therefore, there exist o E and asubsequence k \u00E2\u0080\u0094* 0 such that(2.57) \u00E2\u0080\u0094* u0 weakly in(2.58) A11 \u00E2\u0080\u0094* Auo weakly in1 O\u00C3\u00B1- 1 Onek_(2.59) AIt\u00E2\u0080\u0094 f= \u00E2\u0080\u0094\u00E2\u0080\u0094(\u00E2\u0080\u0094\u00E2\u0080\u0094-\u00E2\u0080\u0094) \u00E2\u0080\u0094 \u00E2\u0080\u0094(\u00E2\u0080\u0094--\u00E2\u0080\u0094-) is bounded in L2.k OX1 Ek Ox2Thus, we have alsoOf \u00E2\u0080\u0094 Oik \u00E2\u0080\u0094(2.60) lim j + (\u00E2\u0080\u0094) j. = 0k\u00E2\u0080\u0094oo Ox1 Ox2i.e.(2.61) urn (\u00E2\u0080\u0094) = 0, i = 1,2.k\u00E2\u0080\u0094oo OxSince the convergence Vi2 \u00E2\u0080\u0094f Vu0 is locally uniform and (.) is a continuous function,we have10u-(2.62) I\u00E2\u0080\u0094} =0, z=1,2,Ox1i.e. (2.56)2 holds for u0. Also, from (2.44) follows A1 f, a.e. in 112; so we can passto the limit over a subsequence Ek to obtain(2.63) Au0 f a.e. in 112i.e. (2.56) holds for u0. Now we assume (xo) > 0 and (x0)> 0 at some x0 E 112then since u0 E we know Vu0 is continuous. Therefore, > 0 (i = 1,2) in aneighbourhood N of x0; again we invoke the locally uniform convergence of V\u00C3\u009C to Vu0and claim\u00E2\u0080\u0094->0 (i = 1,2) in N, fork sufficiently large. So, for such k\u00E2\u0080\u0099s, the equation(2.44) is in fact(2.64) Afi= f a.e in N,Chapter 2. Characterization of the Value Function 33and in the limit as k \u00E2\u0080\u0094* oc, we get(2.65) Aft0= f at x0,but x0 is arbitrary, so (2.56)3 holds for u0. But fr \u00E2\u0080\u0094* ii (cf. Corollary 2.5), hence n0 = ftand (2.56) holds for ft.Finally, we observe that any solution of (2.56) is also a solution of (2.42); therefore,it follows from Theorem 2.9 that ft is the maximal solution of (2.56). \u00E2\u0080\u00A2Remark 2.12 For u e G, the inequality n Mu, a.e. in ]R2 (i.e. u(x) u(x + )for all E A*) is equivalent to Vu 0, a.e. in 112 and for all E A*, and thereforeto (2.56)2 (from the definition of A*). So Theorem 2.9 and Theorem 2.11 are essentiallyequivalent, which is not surprising if one realizes that they both characterize the samefunction ft, but by the two different approaches ft \u00E2\u0080\u0094 ft and ftE ft, respectively.Chapter 3The Free Boundary3.1 IntroductionIn Section 3.2 we identify the free boundary DA\u00E2\u0080\u0099 of the region of inaction A1 and the threeregions into which the region of action (A splits. Then, in Section 3.3, we assume (LFP)and we study the measure theoretic boundary of A\u00E2\u0080\u0099, OMA1 by means of sets of finiteperimeter and their tangential properties in the measure theoretic sense. In Section 3.4we improve the previous results and we obtain the regularity of a new boundary DA\u00E2\u0080\u0099which is \u00E2\u0080\u009Cequivalent\u00E2\u0080\u009D (in the measure theoretic sense) to the free boundary 0A1; in fact,we show that \u00C3\u0094A\u00E2\u0080\u0099 can be replaced by OA\u00E2\u0080\u0099. In Section 3.5 we give conditions under whichthe local finiteness of the perimeter (i.e. (LFP)) of A\u00E2\u0080\u0099 holds. Finally in Section 3.6 weupgrade the regularity of the boundary as well as that of the value function \u00C3\u00BC under theconditions of Section 3.5.3.2 Identification of the BoundaryIn this section a description of the boundary of the region of inaction is obtained as ageneralization of the fact that in the one dimensional case this boundary is exactly thefirst point of strict increase of the value function %. Such a result was already implicitin [MRbj, Theorem 4.1, although there the proof is quite confusing and seems to containseveral gaps. We reformulate and prove a similar result.34Chapter 3. The Free Boundary 35Theorem 3.1 There exist two functions \u00E2\u0080\u0098b2(xi) and \u00E2\u0080\u0098bi(x2) such thatI,2(xi,x)= 0 Vx2 &2(xi),(3.1) Vx1EIR: \u00E2\u0080\u0098(i2(xi,x) > 0 Vx2 >\u00E2\u0080\u00982(x1);I 1(ix) = 0 Vx1 I\u00E2\u0080\u0099i(x2),(3.2) Vx2EIR: c1(rix2) > 0 Vx1 > \u00E2\u0080\u0098i(x2);i.e. the functionsb2(x1) and \u00E2\u0080\u0098\u00C3\u00A7b1(x2) are defined by() Jb2(xi) = inf{x2 :2(x1,)> 0},(1x2)= inf{xi :\u00C3\u00B11(xi,x2)> 0}.Proof: We recall that E C\u00E2\u0080\u0099(]R2) (cf. Theorem 2.3).Claim 1: Vx2, such that Vx1, i21(x,2)> 0.With i E II, and c> 0 (both to be fixed later) we define the following function,10 ifx1w(x1,x2)=( c(xi \u00E2\u0080\u0094i) if x1Since Aw(xi, x2) = \u00E2\u0080\u0094g1c+ pc(xi\u00E2\u0080\u0094for x1,we can choose c> 0, E IR suchthatAw(xi,x2) f(x1,x2) if x1(This can be done because the assumed polynomial growth condition (2.4) impliesr(xi,x2)jP\u00E2\u0080\u0094 C0 f(xi,x2) with C0 and r independent of x2.) Since f 0 thenw is a solution of the quasi-variational inequality (2.42), whose maximal solution is (cf.Theorem 2.9); so we conclude that w i. Therefore, since 0, for every x2 thereexists some point \u00C3\u00A3 i such that \u00E2\u0080\u0098i% (ii, x2) > 0; from this we now deduce Claim 1(since \u00C3\u00B1 is convex).Similarly, we have Vx1 3x2 such that Vx2 : u2(x1,x2) > 0.Claim 2: Vi : x2) remains bounded as x2 \u00E2\u0080\u0094* \u00E2\u0080\u0094oc, (i = 1,2).Chapter 3. The Free Boundary 36In fact, X2(1,\u00E2\u0080\u00A2) is non-decreasing (by convexity) and 0 (by Theorem 2.11). Sou2(x1,)\u00E2\u0080\u0094* cc as x2 - \u00E2\u0080\u0094cc would implyi2(,\u00E2\u0080\u00A2) +oo, which is impossible. Onthe other hand, since ii is convex and nonnegative we have\u00E2\u0080\u0098\u00C3\u00BC(i+h,x2) \u00C3\u00B1(i,x2)+h1(i,hIt1(i,x2) for every x2.Therefore,n1(,x2)\u00E2\u0080\u0094* +00 as x2 \u00E2\u0080\u0094* \u00E2\u0080\u0094cc would imply (for h>0) +h,x2)\u00E2\u0080\u0094* +ccas x2 \u00E2\u0080\u0094* \u00E2\u0080\u0094cc, so \u00C3\u00BC(1 + h,.) +cc (since \u00E2\u0080\u0098\u00C3\u00BC2 0). This is impossible because of thepolynomial growth of .Claim 3: Vx1 2 such that \u00E2\u0080\u0098\u00C3\u00BC2(xi,.) 0.In fact, if not, then we have that such that2(1,x)> 0 for all x2. So one ofthe following two cases occurs:case (a): X2,n \u00E2\u0080\u0094* \u00E2\u0080\u0094cc such that,1(1,x2,fl) > 0;orcase (b): x2 such thatf1(,x2)= 0 Vx2 x2.In case (a), by continuity of I and \u00C3\u00B1a, we have \u00E2\u0080\u0098It > 0 and fL2 > 0 on some open setU containing(1,x2fl). Then, by Theorem 2.11, An= f a.e. in U, i.e.(3.4) \u00E2\u0080\u0094 tr [*_]\u00E2\u0080\u0094 g. V + pn = f a.e. in U.From (3.4) and tr 0 a.e. (since \u00C3\u00BC is convex) follows(3.5) p\u00C3\u00BCf+g\u00E2\u0080\u00A2V\u00C3\u00BC inU,,so(3.6) pn(1,x2 with(1,x2)e U,and in particular(3.7) pn(1,x2,) f(i,x2,) + giii.1(ii,x2,) +92x(ii\u00E2\u0080\u0099x2,).Chapter 3. The Free Boundary 37Because we are assuming f(xi,x2)/I(xi,I \u00E2\u0080\u00944 +00 as I(xi,x2)I \u00E2\u0080\u0094* +oo, we havef(1,x2) \u00E2\u0080\u0094* +00 as n \u00E2\u0080\u0094* +00, whilen1(,x2)and2(1,x2,fl) stay bounded as\u00E2\u0080\u0094* +00 (by Claim 2). Then, from (3.7) we havePU(X1,X2,n) \u00E2\u0080\u0094f +oo as n \u00E2\u0080\u0094> +00,but n(1,.) is non-decreasing, therefore it must be \u00C3\u00B1(1,.) +oo and this contradictsthe polynomial growth of \u00C3\u00B1. So case (a) cannot occur.Let us now assume that case (b) holds. Then, because of convexity and i2 0, wehave(3.8) 1(x,2)= 0 for every (x1,2)e (1,2) \u00E2\u0080\u0094 A*.Now Claim 1 allows us to define, for x2\u00E2\u0080\u0098&1(x2)= max{xi :1(xl,x2) = 0} = inf{xi :1(x,z2)> 0}.WedefinetheregionA = int({(xi,x2): x1 1(x2), x}). Fromthedefinitionoffollows that ii,,, 0 in A. Moreover, by assumption i2 0 on the line x1 = also, onthe left of b1, is constant along horizontal line segments and therefore 2 is constantalong horizontal line segments. In conclusion,2(1(x2),x2) =?2(1,x2) > 0 forx2 x2. So by continuity of ii2 we have \u00E2\u0080\u0098ix2 0 in U,1 fl A, U1 being a neighbourhoodof DA. Then, 0 and i2 0 in U1, fl A. Now Theorem 2.11 implies that thedynamic programming equation holds in U,1 fl A, i.e. \u00E2\u0080\u0094tr {uu*J\u00E2\u0080\u0094 g V + p\u00C3\u00B1 = fa.e. in U,,1 fl A. We know tr {aa*] 0 a.e. (since \u00C3\u00B1 is convex), so we havepi,x2) f(,x2) + gi1(, x2) +g2u(,x2) if (, x2) E U1 fl A.It follows from the continuity of \u00C3\u00B1 and the definition of that 1Iii b1(t) 1(x2),tZ2hence(1x2),x2) E OA for any x2 <.2. Therefore, we can take - urn and obtainz\u00E2\u0080\u0094+i,bj (z2)(z\u00E2\u0080\u0099,T2 )EAflU1Chapter 3. The Free Boundary 38(by continuity)(3.9) p2(j(x),x f(ibi(x2),x+g2i(\u00C3\u00A7&1x),, with x2 <2.Also, i(1,x2)= ui(\u00E2\u0080\u0099bi(x2),x (by the definition of, since (3.8) implies \u00E2\u0080\u0098bi(x2)for every x2 2); \u00E2\u0080\u0098iL2(i,2) n2(1,x2) (by convexity and x2 x2) finally,i2(i(x), =i2(1,x)(as we observed above). Then, from (3.9) followsp(1,x2) f(bi(x2),x+g22(1,x)f(\u00E2\u0080\u0099i(x2),x \u00E2\u0080\u0094Ig2u(x1,).Using the fact thatf(xi, x2)+00 as I(xi,x2)I \u00E2\u0080\u0094 +00we obtain\u00E2\u0080\u0098z2(1,x2)\u00E2\u0080\u0094* +00 as x2 \u00E2\u0080\u0094 \u00E2\u0080\u009400which is impossible since i(1,x2)decreases as x2 \u00E2\u0080\u0094f \u00E2\u0080\u009400 (as i2 > 0). Therefore,case (b) cannot occur and Claim 3 is finally proved.Similarly we have Vx2 such that (ii, x2) = 0. Then, the functions in (3.3)are well defined. RWe remark that Menaldi and Robin (cf. [MRb], Theorem 4.1), claimed that thefunctions \u00E2\u0080\u0098/\u00E2\u0080\u0098j are non-increasing. However, their proof is incorrect. We were able to provethis result only under some extra conditions, e.g. \u00C3\u00B12 0.We now define{(x1,x2): x1 b1(x2), b2(xi)},(3.10) R1 := {(x1,x2): x1 &(x), O,\u00C3\u00BC2(x ,x)> O}is open since \u00E2\u0080\u0098\u00C3\u00BC, and u,2 are continuous; therefore (R0U R, UR2)\u00E2\u0080\u0099- is the region of inaction(i.e. the region where Aft = f a.e. holds), and 0 = OR0 fl OA\u00E2\u0080\u0099. Notice that A\u00E2\u0080\u0099 0, asthis follows from Theorem 3.1 and equation (2.7). Also, since f, ft, il are continuousand the \u00E2\u0080\u009Cdynamic programming equation\u00E2\u0080\u009D All = f holds a.e. in A\u00E2\u0080\u0099, this last equalitycan be interpreted to hold everywhere in A\u00E2\u0080\u0099 and tr [*] can be taken to be definedeverywhere in A\u00E2\u0080\u0099 by continuity.Lemma 3.2 Let = inf{ft(x,,x2): (x,,x2) E 1R2}. Then(i) R0 = {(x,,x2) : Vft(x,,x2)= O} {(x,,x2) : ll(x,,x2)= cro};(ii) VP\u00E2\u0082\u00ACR0, P \u00E2\u0080\u0094 A* C R0.Proof: (i) From the definition of follows = 0 in int(Ro). The result now followsby the continuity of \u00C3\u00BC1 and the convexity of ft.(ii) Let P e R0, then 0 (i = 1,2) impliesVQEPA*.Hence Q e R0 by (i). \u00E2\u0080\u00A2Notice that from the definition of /\u00E2\u0080\u0098j, the convexity of Ii, and the fact that 0follows1V(,,2)ER,: (\u00E2\u0080\u0094oo,E,] x {2} CR,,(3.14)e R2 : {\u00E2\u0080\u0098} X (\u00E2\u0080\u0094OO,\u00C2\u00B12] C R2.Chapter 3. The Free Boundary 40Lemma 3.3(i) In R1 one has \u00C3\u00AA2 = const along horizontal line segments.(ii) In R2 one has i = const along vertical line segments.(iii) In int(Ri) one has i22 = const along a.e. horizontal line segment.(iv) In int(R2) one has = const along a.e. vertical line segment.Proof: Let P= (1,2) E R1, then il = const on (\u00E2\u0080\u0094oo,i] x {2} (by the definition ofso, for every fixed 6> 0, one has(3.15) \u00C3\u00B1(\u00E2\u0080\u00A22+6)\u00E2\u0080\u0094(\u00E2\u0080\u00A2, I in (\u00E2\u0080\u0094oo,i)( (.2)\u00E2\u0080\u0094(.,2\u00E2\u0080\u00946) in (\u00E2\u0080\u0094oo,1),since i2,,1 0. It follows that X2(,2) is constant in (\u00E2\u0080\u0094oo,i]. Similarly (ii) follows.The same arguments prove (iii) and (iv), wherever Ii exists. \u00E2\u0080\u00A2Lemma 3.4 The function is upper semicontinuous (u.s.c.), i = 1,2.Proof: Let us recall that 1(z) = inf{xi :21(xi,z) > 0} is defined for every z E JR andis finite (cf. Theorem 3.1). Let z e IR, e> 0, then&i(z)\u00E2\u0080\u0094b1(y) >\u00E2\u0080\u0094E\u00E2\u0080\u0094\u00E2\u0080\u0099&(y)for some such that (, z) > 0, and this holds for any y \u00E2\u0082\u00AC JR. Now from the continuityof \u00C3\u00B1 follows (, y) > 0 if I \u00E2\u0080\u0094 zI <6, for some 6 > 0. Therefore,if I \u00E2\u0080\u0094 zI <6,i.e. is u.s.c.. \u00E2\u0080\u00A2Notice that Lemma 3.4 implies that R0 U R is closed (i = 1, 2).Chapter 3. The Free Boundary 41For a function h 1R2 \u00E2\u0080\u0094* JR and a point Po E 1R2, we set(3.16) h(P0\u00E2\u0080\u0094) := urn h(P), h(Po+) := urn h(P),P\u00E2\u0080\u0094Po P\u00E2\u0080\u0094PoPER0 PEA\u00E2\u0080\u0099if these limits exist. Then we have the followingProposition 3.5(i) Do C {(xi,x2) : f(xi,x2)= Pco} fl {(xi,x2) (x1,2)=(ii) P\u00E2\u0080\u0099o = Ai(Fo\u00E2\u0080\u0094) = = f(P0) for every P0 E Do;(iii) OD(R0uR1)ncD.Proof: Let P0 E 0o, then P0 E R0 and so ui(Po) = a, according to Lemma 3.2. Moreover,since ft is constant in R0 and ft,,,, 2 are continuous, we havePao = pft(Po) = Aui(P0-) f(Po).On the other hand, since ft is convex, tr [u*4(Po+)] 0; therefore,f(P0) = Aft(P0+) = \u00E2\u0080\u0094tr [uu*-(Po+)] + pft(Po) pft(Po).Thus,Aft(Po+) = f(P0) = pft(Po)= Po =Now we show that D(Ro U R1) and D(Ro U R2) intersect. Suppose not, then (sinceis a function of x3, i j) there are essentially three cases (see Figure 3.1):(a) P1 = (xf,x) E O(Ro U Ri) such thatf P2 = (x,x) e D(Ro U R) fl ((\u00E2\u0080\u0094oo,xI) x {x}),> b2(z) for some z x.Chapter 3. The Free Boundary 42(b) Q, = (xl, z4) e O(R0 U R) such thatf = (x,.x) E O(Ro U R,) fl ({x} x (_oo,x)),> ,(z) for some z x.VP1 = (x,x) E O(R0 U R,) : O(Ro U J?2) n ((\u00E2\u0080\u0094cc,xi) x {x}) = 0,(c)lVQ,=(y,y)EO(RoUR2):O(RoURl)fl({y}x(_ o,y)) =0.Notice that (c) is the negation of (a) and (b) under the assumption that O(R0 U R,) flO(R0 U R2) = 0, since \u00E2\u0080\u0098b, and b2 are functions of different variables. Also, (c) occurs ifand only if O(R0 U R,) is to the left and above O(R0 U R2).Let us assume that (a) holds. Then, Vi(P2) = 0 by (3.10) (recall that R0 U R areclosed); so P2 e R0. Also, i(P2) = \u00C3\u00BC(P,) (by the definition of iJ,) since x \u00E2\u0080\u00982(x); i.e. for each n E IN,=0,n2(x,x) >0,asn\u00E2\u0080\u0094*oo.From All= f a.e. in A\u00E2\u0080\u0099 and tr {* 0 follows\u00E2\u0080\u0094g.Vll+puif mA\u00E2\u0080\u0099(recall that ii, f ll, are continuous). Hence, by continuity, the same is true 011graph(,), and so we havef(x,x), for n E IN.Chapter 3. The Free Boundary 43Figure 3.1: Possible cases (a), (b) and (c) occurring in the proof of Proposition 3.5,taking 1, 2 continuous.Chapter 3. The Free Boundary 44Now we recall that f(x\u00E2\u0080\u0099,x2)\u00E2\u0080\u0094* +oo as I(x1,x2)I \u00E2\u0080\u0094* oo and so we get(3.17) \u00E2\u0080\u0094glts(x,x)+p11(x,x)\u00E2\u0080\u0094* +00 as n \u00E2\u0080\u0094* +00.However, 11(x, x) \u00E2\u0080\u0094* +00 as n \u00E2\u0080\u0094+ oo would imply \u00C3\u00B1 +oo since 12 is convex (and0, i%a2 0) and since x \u00E2\u0080\u0094* \u00E2\u0080\u0094oo. Then 12(x,x) \u00E2\u0080\u0094* +oo is impossible. Similarly,122(x,x) \u00E2\u0080\u0094* +00 as n \u00E2\u0080\u0094* oo cannot hold for that would imply \u00E2\u0080\u0098u2 +00 on graph(i&1);in fact, 122(x,.) I (by convexity) and \u00E2\u0080\u0098iL2 const along horizontal line segments tothe left of graph(i) (by Lemma 3.3(i)) imply ltx2(&1(.),.) I. Clearly 12x2 +00 ongraph(i&i) contradicts the polynomial growth of u2. Hence (3.17) is false and we mustconclude that 0(R U Ri) fl 0(R U R2) 0.Now let P = (xi,x2) 0(Ro U Ri) fl 0(1?o U R2), thus P E 0(R U R1 U R2), andP e 0(R). Hence P E 0o (ef. (3.12)). \u00E2\u0080\u00A2Proposition 3.6 oo is a singleton.Proof: Let B = {P : f(P) b2(xi). Consider the linesegment S = {(x1,x2): x1 = x2}. Then \u00E2\u0080\u0098z2 is constant on S (since \u00C3\u00B1 = 0 on Sby the definition of b1). If SflR0 0, then fi = o on S (cf. Lemma 3.2) and so P E R0,which is impossible because 2 > &2(x1). Therefore, Sn R0 = 0, i.e. R0 has to lie belowany horizontal line through 01 \ 00. Similarly R0 must lie to the left of any vertical linethrough 02 \ 00. Since 01 and 02 meet at 00 we have R0 C 0o \u00E2\u0080\u0094 A*. Now the assertions ofthe Proposition follow from the definition of R0, Lemma 3.2 and Proposition 3.6. \u00E2\u0080\u00A2Lemma 3.8 If x, x\u00C2\u00B0 are defined as in Proposition 3.7, then/jis constant on (\u00E2\u0080\u0094oo,x9,i j, i,j = 1,2.Proof: Certainly (x) = \u00E2\u0080\u0098b(x) on (\u00E2\u0080\u0094oo, x], i j, since R0 = 0 \u00E2\u0080\u0094 A* (see Proposition 3.7).Lemma 3.9 The function is continuous at x3\u00C2\u00B0 , i j (i,j = 1,2). Hence, in particular,0 graph(1)fl graph() = oo = {Po}.Proof: Let us fix i = 1, j = 2 for simplicity. By Lemma 3.4, bi is u.s.c.; therefore,if bi(x+) := lim 1(z), we have 1(x+) \u00E2\u0080\u0098\u00C3\u00A7b(x). On the other hand, since x =z\u00E2\u0080\u0094xo= tb2(x1) for any x1 E (\u00E2\u0080\u0094oo,x?) (cf. Proposition 3.7 and Lemma 3.8),1(x+) <1(x) would imply x {x} C 0 (cf. (3.12) for the definition of 0w),but this is impossible since 0 is a singleton (cf. Proposition 3.6). So1(x+)=also, \u00E2\u0080\u0098\u00E2\u0080\u00981(x\u00E2\u0080\u0094) = 1(x) by Lemma 3.8. The last assertion of the Lemma follows fromPropositions 3.5\u00E2\u0080\u00943.7. \u00E2\u0080\u00A2Hence we have established that the functions j cause the plane to split into the fourregions R0, R1, R2 and A1, as shown in Figure 3.2.Chapter 3. The Free Boundary 46Figure 3.2: Sketch of the region of inaction A\u00E2\u0080\u0099 and its complement R0 U R, U R2.Chapter 3. The Free Boundary 47Remark 3.10 We can show that Vf(P0) points into _A*. This is the 2-dimensionalcounterpart of the fact that in the 1-dimensional case f(xo) 0 if {xo} = 0A1. Toprove this Remark we proceed by contradiction. So if Vf(P0) (A\u00E2\u0080\u009D) then, for example,fx (F0) > 0, and by continuity of f1 we may claim f(P) 0 such that(3.19) fu(x)\u00C2\u00B0) dx t} is of locally finiteperimeter for almost all t (cf. [Zm], Theorem 5.4.4, p.231). Then, there exists a countableset Q, dense in IR such that {n > t} is of locally finite perimeter for all t E Q. Let{tk}1 be a sequence in Q such that tk .t 0 as k \u00E2\u0080\u0094 oo; so we have1{.>tk} E BV0(1R2)for all k E IN; moreover, 1I{%x.>tk} I {>O} as k \u00E2\u0080\u0094* cc. Therefore,i- Il{=o} as k \u00E2\u0080\u0094* ccand{tk} E BV10(IR2) for all k E INChapter 3. The Free Boundary 53imply= 0}, Q) = sup{J 11{.=o}(x) dx : E C\u00C2\u00B0(Q),I(x)I2<1, forxEc}mSUP{J1{Itk}(x)0dx : E C),1, forxEc}= P({1 tk}JZ)k\u00E2\u0080\u0094.oofor all bounded open sets c 1R2. This shows thaturn P({n t,},f) <00 \u00E2\u0080\u0098 (LFP).k\u00E2\u0080\u0094*ooDefinition 3.20 If E C 112 is a Lebesgue measurable set, the measure-theoretic boundary of E is defined by(3.29) OME = {x : D(E,x) > 0} fl {x : D(Ec,x) > 0},where(3.30) D(E, x) = IEflB(x,r)II I being the Lebesgue measure in 112, and B(x, r) being the open ball with center x andradius r. (If llii = urn, we denote their common value by D(E, x).)Remark 3.21 If one says that E is open in the density topology if D(E, x) = 1 for allx E, then it is possible to show that these open sets produce a topology, and OMEis the boundary of E in this topology. (If the exterior of E is the set of all points xsuch that D(E, x) = 0, then OME is the set of all points x which are in neither themeasure-theoretic interior nor exterior of E) (cf. [Zm], Exercise 5.3). Clearly, OME is asubset of the topological boundary OE.Chapter 3. The Free Boundary 54Lemma 3.22 If E C 112 is a Lebesgue measurable set, then the subset of points of OEwhere a tangent exists is contained in the measure-theoretic boundary OME.Proof: Let P E DE be a point where the tangent exists and let n(P, E) denote theoutward unit normal to E at P. Then we have(a) lim 4B(P, r) fl {Q : (Q - P) n(P, E) <0, Q E}I =0r\u00E2\u0080\u0094*O r(b) r) fl {Q (Q - F) n(P, E) >0, Q e E}I =0.Now we define the half-spacesH-(P)= {Q : n(P, E) . (Q \u00E2\u0080\u0094 F) <0},H(P)={Q:n(P,E).(Q\u00E2\u0080\u0094F) >0},then,(a)=-B(P, r) fl (E)C fl H(P) = 0,(b) = lim -B(P, r) fl E fl H+(P)I = 0.Therefore, fromB(P, r) fl H(P) = (B(P, r) fl (E)C fl H(P)) u (B(P, r) fl E fl H(F))followsIB(P,r)flEI IB(P,r) flEflH(P)Ilim > hmrr2 \u00E2\u0080\u0094 r\u00E2\u0080\u0094O irr2\u00E2\u0080\u0094 .IB(P, r) fl H(P)I \u00E2\u0080\u0094\u00E2\u0080\u00987rr2 ISimilarly, we obtainurnIB(P, r) fl (E)dI>1irr2and hence\u00E2\u0080\u00A2 IB(P, r) fl El \u00E2\u0080\u00A2 IB(P, r) fl (E)\u00E2\u0080\u0099l 1hm = urn = -.r\u00E2\u0080\u0094O irr2 r\u00E2\u0080\u0094*O irr2 2\u00E2\u0080\u0099thus, P E 0MB.Chapter 3. The Free Boundary 55Definition 3.23 A subset A of 1R2 is countably 1-rectifiable if(3.31) A c (0 f(A)) U Awhere H\u00E2\u0080\u0099(A0)= 0 and f3:A3 \u00E2\u0080\u0094* 112, A C 11, is a countable collection of Lipschitz maps(H\u00E2\u0080\u0099 being the 1-dimensional Hausdorif measure on 1R2).Remark 3.24 Notice that f can be assumed to be the restriction to A of a Lipschitzmap f3:]R 112, since a Lipschitz map defined on a subset of 11 can be extended to 11by an application of the Whitney extension theorem (cf. [Zm], Theorem 3.6.2).Remark 3.25 A definition of countably 1-rectifiability equivalent to the one above isobtained as a consequence of the C\u00E2\u0080\u0099-approximation theorem for Lipschitz maps (againbased on the- Whitney extension theorem) and Rademacher\u00E2\u0080\u0099s theorem; i.e. we haveA C 112 is countably 1-rectifiable if(3.32)AcUMUNi=1where H1(N) = 0 and each M is a 1-dimensional embedded C\u00E2\u0080\u0099 submanifold of 112 (H\u00E2\u0080\u0099being the 1-dimensional Hausdorif measure on 112). (See [Zm], Lemma 3.7.2, for theproof of this.)Remark 3.26 A fundamental result of De Giorgi shows that, if E is a set of locally finiteperimeter, then its measure-theoretic boundary OME is \u00E2\u0080\u009Cequivalent\u00E2\u0080\u009D to 0E, a subset ofOE having tangential properties in the measure-theoretic sense. In fact, 0E is calledthe reduced boundary of E, and satisfies(a) H\u00E2\u0080\u0099(OME \ 0E) = 0,(b) VP e ThE, a unit vector n(P, E) such that (a) and (b) of Lemma 3.22 hold,(c) ThE is countably 1-rectifiable.Chapter 3. The Free Boundary 56(cf. [Zm], Corollary 5.6.8, Lemma 5.9.5 and Theorem 5.7.3 for the proof of (a)\u00E2\u0080\u0094(c).) Herewe just point out that the proof of part (b) consists in showing that for each x e 0\u00E2\u0080\u0094Eone can find a coneC(x,,v) = {P e R2: (P \u00E2\u0080\u0094 x) v > IP \u00E2\u0080\u0094 xI},with vertex x and major axis parallel to the unit vector v(x), such that C(x, , v) n(0\u00E2\u0080\u0094E) fl B(x, r) = 0 (for some r > 0), with the vector valued function v(x) uniformlycontinuous on compacta in (0E) fl B(x, r).We recall (cf. (3.10)\u00E2\u0080\u0094(3.12)) that(3.33) A\u00E2\u0080\u0099 = (R0 U R U R2)c = {(x,, x2) : x1 > ,(x2),x2 >2(x,)};(3.34) R0 = {(x,,x2) : x1 &,(x2),x \u00E2\u0080\u0098&(x,)} = {n = 0 =(3.35) R, = {(x,,x2) x1 \u00E2\u0080\u0098b,(x2),x > \u00E2\u0080\u0098b(x,)} = = 0iiX2 > 0};(3.36) R2 = {(x,,x2) : x1 > \u00E2\u0080\u0098bi(x2),x \u00E2\u0080\u0098b(x,)} = {\u00C3\u00B1 > 0,ii2 = 0}.Since the sets {I = 0} are assumed to be of locally finite perimeter, the remarks aboveallow us to claim regularity for H\u00E2\u0080\u0099-almost all of OM{ = 0}. Also, we observe that theunion of two sets of locally finite perimeter is a set of locally finite perimeter (cf. [Gs],Remark 1.7, pM). Then, since OA\u00E2\u0080\u0099 = 0, U O (see Theorem 3.11) where 0 is a connectedsubset of 0{i, = 0}, we obtain the followingTheorem 3.27 Assume (LFP). Then(i) the region of inaction A\u00E2\u0080\u0099 is of locally finite perimeter;(ii) DMA\u00E2\u0080\u0099 is countably 1-rectifiable, i.e. DMA\u00E2\u0080\u0099 C U N where\u00E2\u0080\u00A2 H\u00E2\u0080\u0099(N) = 0 (H\u00E2\u0080\u0099 being the 1-dimensional Hausdorff measure on 1R2),\u00E2\u0080\u00A2 each M1 is a 1-dimensional embedded C\u00E2\u0080\u0099 submanifold of 1R2.Chapter 3. The Free Boundary 57Now the question is, how \u00E2\u0080\u009Cbig\u00E2\u0080\u009D is DA\u00E2\u0080\u0099 \ DMA\u00E2\u0080\u0099? In the next section we will show thatit is possible to redefine A\u00E2\u0080\u0099 in order to obtain a new region of inaction A\u00E2\u0080\u0099 such that DA\u00E2\u0080\u0099equals DMA\u00E2\u0080\u0099 at least up to sets of lower dimension.3.4 Regularity of the Free BoundaryIn order to answer the question posed at the end of the previous section we must analyzethe free boundary DA\u00E2\u0080\u0099 in more detail and see how many \u00E2\u0080\u009Cnasty\u00E2\u0080\u009D points are in it. In fact,since (as we have seen in Lemma 3.22) all points of DA1 where a tangent vector exists arealso points of the measure-theoretic boundary DMA\u00E2\u0080\u0099, it is natural to start the study ofDA\u00E2\u0080\u0099 \ DMA\u00E2\u0080\u0099 by examining those points of the boundary where the tangent fails to exist.Fortunately we have already obtained a parametric representation of A\u00E2\u0080\u0099 (cf. (3.33)) interms of the functions b,, &2 so that the problem is now reduced to the study of thedifferentiability of b, and I\u00E2\u0080\u00992. We begin with some properties of \u00E2\u0080\u0098/-\u00E2\u0080\u0098,, b2 which will lead usto the existence of two functions, and 2, differentiable almost everywhere and suchthat /\u00E2\u0080\u0098 =/,ja.e., i = 1,2.Lemma 3.28 The function \u00E2\u0080\u0098j is locally bounded (i = 1,2).Proof: Fix i = 1. Recall that is u.s.c. (cf. Lemma 3.4), so b, is bounded aboveon compacta. Hence it suffices to show that b, is bounded below on any compact setK C [x, +oo) (cf. Lemma 3.8). Assume not, then e K such that urn \u00E2\u0080\u0098\u00C3\u00A7b,(y) = \u00E2\u0080\u0094oc;v\u00E2\u0080\u0094*z1yE Kso there exists a sequence Yn \u00E2\u0080\u0094 z1 such that y, E K and \u00E2\u0080\u0098hI\u00E2\u0080\u0099i(Yn) \u00E2\u0080\u0094 \u00E2\u0080\u0094oc as n \u00E2\u0080\u0094 oc. Also,since K c [x, +oc) we have{(\u00E2\u0080\u0098!-\u00E2\u0080\u0098i(Yn),Yn) : n E IN} cD1.By Theorem 2.11, Ai = f a.e. in A\u00E2\u0080\u0099, but tr [uu*4] 0 a.e. (since Ii is convex), hencepu f + g Vi2 a.e. in A\u00E2\u0080\u0099Chapter 3. The Free Boundary 58and by continuity this inequality can be interpreted to hold everywhere in cl(A\u00E2\u0080\u0099). Inparticular, one has (cf. Theorem 3.11)pii\u00E2\u0080\u0094g2il>f on\u00C3\u00B61.Therefore,fQtLi(y),y) pit(i/\u00E2\u0080\u0099i(y),y)\u00E2\u0080\u0094g2Ux(/)i(yn),yn),but then, in the limit as n \u00E2\u0080\u0094* 00, the LHS will diverge to +00 since\u00E2\u0080\u0094\u00C3\u00B7 +00 asI(xi,x2)I \u00E2\u0080\u0094+ +00, while the RHS will remain bounded since(\u00E2\u0080\u0098i(yn),yn) (\u00E2\u0080\u0098bi(Zi),Yn) \u00E2\u0080\u0094*i1x2(\u00E2\u0080\u0099I-\u00E2\u0080\u0099i(Yn),Yn) fLx2(\u00E2\u0080\u0099L\u00E2\u0080\u0099i(yn),Zi) =\u00C3\u00B12(bi(zi),zi) if y 1\u00E2\u0080\u0099 z1,\u00E2\u0080\u009822 (u/i(y), y,) is non-increasing with n if y J z1as follows from the fact that in R1 both \u00E2\u0080\u0098t and \u00C3\u00B12 are constant along horizontal linesegments (cf. Lemma 3.3), and \u00C3\u00B1 0. Contradiction! (For\u00E2\u0080\u0098\u00E2\u0080\u00982 the proof is thesame). \u00E2\u0080\u00A2Remark 3.29 It is known that the set of points of discontinuity of the first kind of a realfunction iJ\u00E2\u0080\u0099: JR \u00E2\u0080\u0094 JR is at most countable. Therefore, and /\u00E2\u0080\u00982 have at most a countablenumber of jumps, and to each one of them there corresponds a finite line segment onthe free boundary DA\u00E2\u0080\u0099. In order to guess the behaviour of the optimal process at theboundary of the region of inaction and then be able to construct the optimal control ofthe problem (2.3) we need to know what the boundary looks like. If the functions b,and\u00E2\u0080\u00982 have discontinuities of the second kind, then it becomes impossible to picture theshape of the boundary near those points and then discover the optimal strategy. That iswhy we would like to exclude the possibility of points of discontinuity of the second kindfor \u00E2\u0080\u0098L\u00E2\u0080\u0099, and 2; as we have been unable to do so, we try to circumvent the problem byChapter 3. The Free Boundary 59showing that the set of all such points has measure zero, hence the region of inaction canbe slightly modified by changing the boundary on a set of measure zero without affectingthe value function i and its properties at all.Throughout the remainder of this section we assume that (LFP) holds. Then, the setR0UR1 (i = 1,2) is a set of locally finite perimeter (cf. (3.34)\u00E2\u0080\u0094(3.36)), hence the gradientof\u00E2\u0080\u0098ROUR is a vector valued measure whose total variation over any bounded open seto C JR2 is finite (cf. (3.20) and (3.24)) and given byIID1uRII(O) = sup{f \u00C3\u00B4) dx : j E C\u00C2\u00B0(O),Ij(x)I21, forxEO}.j=1Now we extend this definition to sets of the form JR x ! and f x JR where is a boundedopen set in II, i.e.IIDhhi?ouRi II( x = sup{JR0U1Oi(x)dx : E C\u00C2\u00B0(JR x1)12, forxeJRxc}IID1&uR2I( x R) = sup{J dx: j E C( x H),I(x)I21, forxE1.xJR}(3.37)where is a bounded open set in JR.In general, if E is a set of locally finite perimeter in H2, the total variation JIDIIEII(lRx1) need not be finite as JR x is not bounded. However, the set R0 U R is the closure ofthe subgraph of the function \u00E2\u0080\u0098\u00C3\u00A7b (cf. (3.10)) and is locally bounded (cf. Lemma 3.28),so it is then clear that IID1JrouRi II(IR x ) < oc if is bounded and open (the sameconclusion holds for R0 U R2 m\u00E2\u0080\u0099utatis mutandis). This fact is here stated asProposition 3.30 If (LFP) holds and Q is a bounded open set in H, thenChapter 3. The Free Boundary 60(i) IID1IuR1I( R x ) <00;(ii) IID1IuR2I( x IR) <00.Proof: Let be a bounded open set in IR, then it follows from Lemma 3.28 that thereexists T> 0 such that(\u00E2\u0080\u0094oo,\u00E2\u0080\u0094T) x f C ((RouRi)n(lRx ci)) C (\u00E2\u0080\u0094oo,T) xci;therefore,ID1uRiII(R x = IID1&uRiII((\u00E2\u0080\u0094T,T) xci),and this is finite since R0 U R is a set of locally finite perimeter, hence (i) is proved. Theproof of (ii) is the same. \u00E2\u0080\u00A2For the sake of completeness we report here some of the properties of the sets havingfinite perimeter in IR x ci.Remark 3.31 It was shown by Miranda that (cf. [MrJ, Teorema 2.1, p.526) if (9 is anopen connected subset of IR and E is a measurable subset of H x (9, such that(3.38) IDlE II(1R x ci) <00for every bounded open set ci, cl(ci) C 0, then E must necessarily satisfy the followingalternatives10 fora.e. x2EO(3.39) lim IIE(X1,X2) = orI.. 1 fora.e. x2O,10 fora.e. x2O(3.40) lim 1E(x1,X2) = or1 1 fora.e. x2EO.In particular, if E satisfies (3.38), (3.39a) and (3.40b), then (cf. [Mr], Teorema2.3, p.531)E L1\u00E2\u0080\u00990(O) such that if L = {(x1,x2)e H x (9: x1Chapter 3. The Free Boundary 61one has(i) J11E(x1,x2)\u00E2\u0080\u0094 1L(xl,x2)I dx1 e L0(Q)and(ii) fdx2J[I1E(xl,x)\u00E2\u0080\u0094 IIL(xl,x2)j dx1 = 0for every bounded open set , cl() C (9.That is, all sets E satisfying (3.38) and { (3.39a), (3.40b) } or { (3.39b), (3.40a) } are\u00E2\u0080\u009Cessentially\u00E2\u0080\u009D \u00E2\u0080\u009Cequivalent\u00E2\u0080\u009D to sets of the form L, i.e. subgraphs of functions \u00E2\u0080\u0098\u00C3\u00A7& E Lj0(O)(here \u00E2\u0080\u009Cequivalent\u00E2\u0080\u009D is in the sense of (i), (ii) above, and \u00E2\u0080\u009Cessentially\u00E2\u0080\u009D means that eitherE or its complement E\u00E2\u0080\u0099 is the subgraph of some e Lj0(O)).All the information so far collected allows us to obtain some regularity of the functionsi, \u00E2\u0080\u0098\u00C3\u00A7b2 by applying a result of Miranda which connects functions b BV0(R) andsubgraphs L of locally finite perimeter.Definition 3.32 Let 0 be an open set in IR and E BV(0). Then the set {(x,y)x e 0, y = (x)} is called a generalized cartesian curve, and for every bounded open setC 0, cl() C 0, we say the length of the portion of the curve defined on Q is the totalvariation of the vector valued measure (D\u00C3\u00A7, )) over and we denote it by j/i + IDI2.Hence(3.41) J 1 + ID := suP{L(2(x) + (x)(x)) dx : E C), E II2 i}.(Here A is the Lebesgue measure in IR.)Remark 3.33 As in Remark 3.14, if \u00C3\u00A7 e W\u00E2\u0080\u009D(12) we have (cf. [Gsj, p.160)j 1 + IDI2 = j 1 + II(x)I2 dx.Chapter 3. The Free Boundary 62Theorem 3.34 Under the assumption (LFP), the functions \u00E2\u0080\u0098\u00E2\u0080\u0098i and 2 enjoy the following properties:(3.42) /\u00E2\u0080\u0098, E BV0(1R), i = 1,2.I IID1JuRiII(1R x= J V/i + D&iI2,(3.43)_________[ !IDu1RouR2I( x II)= f V/i + ID2;(344) f IIDuil 11(11 x = IIDiII(I IIDiu2II( x H) = IID2();J lID\u00E2\u0080\u00991iRuR, 11(11 x =I IID2&uR2lI( x IR) =for every Borel set C IR.(Here IID1EII(O) is the total variation of the i-th component D2IIE of the vectorvalued measure DuE over the set 0; )() is the Lebesgue measure of in H.)Proof: Because of (LFP) and Proposition 3.30 we can apply Teorema 1.10 in [Mr}, p.525,and obtain (3.42)\u00E2\u0080\u0094(3.45). \u00E2\u0080\u00A2In particular, under the assumption (LFP), the previous Theorem provides us withE B40(1R), i = 1,2, but then (cf. Remark 3.15(c)) there exists a function iJ\u00E2\u0080\u0099j of locallybounded variation in the usual sense, such that/ij= a.e. We are going to show(I) the boundary O has 2-dimensional Lebesgue measure zero, i.e. IOZ = 0, i = 1,2;(II) the dynamic programming equation \u00E2\u0080\u009CAi= f a.e. in A\u00E2\u0080\u009D can be interpreted to holdacross the non-dense part of the region of inaction, i.e. \u00E2\u0080\u009CAu= f a.e. in int(cl(A\u00E2\u0080\u0099))\u00E2\u0080\u009D;(III) the region of action A\u00E2\u0080\u0099 can be redefined as A\u00E2\u0080\u0099 = int(cl(A\u00E2\u0080\u0099)), and there is a uniquechoice of,b,, b2 such that A1 = {(x,,x2) : x1 > ,(x2), >Chapter 3. The Free Boundary 63(IV) H\u00E2\u0080\u0099(OA\u00E2\u0080\u0099 \ 9MA) = 0;(V) DA\u00E2\u0080\u0099 is countably 1-rectifiable (cf Definition 3.23).In order to prove (I) we need to show that the set of points of discontinuity of the 2ndkind of has zero Lebesgue measure, so that (I) follows from Fubini\u00E2\u0080\u0099s theorem andRemark 3.29.Theorem 3.35 If L E BV10(R) is upper semicontinuous and D is the set of all itspoints of discontinuity then= 0.Proof: It suffices to show that the set of points where \u00E2\u0080\u0098/\u00E2\u0080\u0098l[a,b] is discontinuous has measurezero for all finite intervals [a, b]. So let us fix a < b and let b E BV([a, b]); we again set(3.46) D1, = {x e [a, bj : x is a point of discontinuity of }.Assume A(D,) > 0. Also, e BV([a, b]) and Remark 3.15, (c) imply the existenceof a function of bounded variation (i.e. V() < oc) such that & = a.e. in [a,b].Therefore, since b may have at most countably many discontinuities,on [a, b] \ N(3.47) N C [a, b] such that is continuous on [a, b] \ NA(N) = 0.Thus, A(D, \ N) > 0 and we set(3.48) D\u00E2\u0080\u0099 = D \ N;moreover, we may assume the elements of D1, to be points of discontinuity of the 2ndkind, since those of the 1st kind are at most countable (cf. Remark 3.29) and hence maybe assumed to be elements of N.Chapter 3. The Free Boundary 64Let y e D. Let 6 > 0 be fixed and such that 8 < \u00E2\u0080\u0098b(y) \u00E2\u0080\u0094 lim\u00E2\u0080\u0099b(z). This is possiblesince b is u.s.c. and we are assuming that y is a point of discontinuity of the 2nd kind,so \u00E2\u0080\u0098(y) flm\u00E2\u0080\u0099b(z) > lim\u00E2\u0080\u0099/\u00E2\u0080\u0099(z). By continuity, since y e [a,b] \ N, there exists no E INysuch that(3.49) Iz\u00E2\u0080\u0094 I <1/no, z e [a, bj \ N = kb(z) \u00E2\u0080\u0094 \u00E2\u0080\u0098\u00C3\u00A7t\u00E2\u0080\u0099(y) <.On the other hand, since \u00E2\u0080\u0098(y) \u00E2\u0080\u0094 e> lim\u00E2\u0080\u0099b(z), it follows from (3.49) thatz\u00E2\u0080\u0094*y(3.50) xo E N fl B(y, 1/no) such that (x0) 0: B(Po,r)flRoI =that is,P0 int(cl(A\u00E2\u0080\u0099)) = A\u00E2\u0080\u0099,but {Po}=OoCOA\u00E2\u0080\u0099,soitmustbePo. \u00E2\u0080\u00A2Proposition 3.39 Assume (LFP). Then R has zero Lebesgue density at every point ofO \O, i = 1,2; i.e.(3.61) VP e O \ 0 : D(R, F) = 0, i = 1,2.Chapter 3. The Free Boundary 67Proof: We recall that(3.62) D(R1,F) \u00E2\u0080\u0094 lirn IRjflB(P,r)IFix i = 1 for simplicity and let P E O \ a,, then P \u00E2\u0082\u00AC A\u00E2\u0080\u0099 (by Lemma 3.38) and A\u00E2\u0080\u0099 isopen, so B(P, r) C A\u00E2\u0080\u0099 for r sufficiently small, henceIR,flB(P,r)I IR,flA\u00E2\u0080\u0099I IO,I,but \u00E2\u0080\u0098,I = 0 by Proposition 3.37, thus (3.61) follows. \u00E2\u0080\u00A2Corollary 3.40 Assume (LFP). Then the set \u00C3\u0094A\u00E2\u0080\u0099 \ DA\u00E2\u0080\u0099 has empty intersection with themeasure theoretic boundary of A\u00E2\u0080\u0099, OMA\u00E2\u0080\u0099, i.e.(a.6a) OMA C aA\u00E2\u0080\u0099.Proof: This follows immediately from Proposition 3.39 and the fact that all the elementsof 9MA\u00E2\u0080\u0099 are points of positive Lebesgue density for both A\u00E2\u0080\u0099 and its complement (cf.Definition 3.20). \u00E2\u0080\u00A2We have shown that 0 \ 6 is a subset of int(cl(A\u00E2\u0080\u0099)) but its 2-dimensional measureis zero. It is then easy to convince ourselves that, after all, 0 \ 0 is a \u00E2\u0080\u009Cfalse\u00E2\u0080\u009D region ofaction, since adding 0 \ 0 to A\u00E2\u0080\u0099 does not affect the dynamic programming equation; i.e.(II) must hold. In fact, from Lemma 3.38 we haveA\u00E2\u0080\u0099 =A\u00E2\u0080\u0099n(R1uR2\u00E2\u0080\u0099)= (OA\u00E2\u0080\u0099 \ aA\u00E2\u0080\u0099) U (A\u00E2\u0080\u0099 n A\u00E2\u0080\u0099),but A\u00E2\u0080\u0099 c int(cl(A\u00E2\u0080\u0099)) = A\u00E2\u0080\u0099, hence(3.64) A\u00E2\u0080\u0099 = A\u00E2\u0080\u0099 u (aA\u00E2\u0080\u0099 \ aA\u00E2\u0080\u0099).Chapter 3. The Free Boundary 68Therefore, (cf. (2.47), (3.56))(3.65) All= f a.e. in A1,(3.66)together with (3.64), imply (II), i.e.(3.67) All= f a.e. in A\u00E2\u0080\u0099.As we are planning to redefine the region of action and work with A\u00E2\u0080\u0099 instead of A\u00E2\u0080\u0099,we want to obtain a representation of A\u00E2\u0080\u0099 similar to the one provided by (3.33) for A\u00E2\u0080\u0099,i.e. we will show(3.68) A\u00E2\u0080\u0099 = {(x,,x2)e 1R2 : x > ,(x2), > (xi)}where \u00E2\u0080\u0098i, \u00E2\u0080\u0098\u00E2\u0080\u00982 can be selected to be u.s.c. (just as b,, \u00E2\u0080\u0098b2 were u.s.c.) and of locallybounded variation in the usual sense (instead, b,, \u00E2\u0080\u0098\u00C3\u00A7b2 were only elements of BV0(1R)).This is what point (III) is about.We start by showing that,,\u00E2\u0080\u0098\u00C3\u00A7b2 can be uniquely chosen to be u.s.c. among all thefunctions provided by Remark 3.15, (c).We set(3.69) bv(b) = {b:1R \u00E2\u0080\u0094 IR=a.e., V@) 0 6o > 0 such that 0 < Iz \u00E2\u0080\u0094 xI 0 o> 0 such that 0 < Iz \u00E2\u0080\u0094 xI <6 (z) b(x) + a,therefore(3.73) rn(z) (x),i.e. e {u.s.c.}.Chapter 3. The Free Boundary 70We now show that q is of locally bounded variation; in fact, if a, b E R are points ofcontinuity of such that a 0, then from= ] 7()follows(i) 6j > 0 such that 0 < I \u00E2\u0080\u0094 x <6 = \u00C3\u00AD) (x) +and with 6 < min{6i, 62,... , \u00C3\u00B6\u00E2\u0080\u0094i,(ii) z2, ..., z_1 such that 0 < Iz \u00E2\u0080\u0094 Xj <6 and (z) (x) \u00E2\u0080\u0094So if we set z0 = a, z = b we obtain a new partition a = z0 < Zi < < Zn_i < Zn = bof [a, b] such that(xj)\u00E2\u0080\u0094(z1)I ifi= 1,2,...,n\u00E2\u0080\u00941,andI(x) \u00E2\u0080\u0094 cb(xj)I = I(x) \u00E2\u0080\u0094 (x)1 = 0 if j = 0, nsince a, b are points of continuity of , hence \u00C3\u00A7 equals q there. Then,Z I(x) - (x)I{I(x) - (Z)I + I(z) - (Z)I + I(x)-+ qi) \u00E2\u0080\u0094 q(Zi_1)I +E + I(z) \u00E2\u0080\u0094i=iChapter 3. The Free Boundary 71therefore, V() + Vj\u00E2\u0080\u0099(), and since \u00E2\u0082\u00AC > 0 is arbitrary (3.74) follows. Thus,for every a, b E IR, a< b(if a, b are points of discontinuity of q, then we can always find a\u00E2\u0080\u0099 < a and b\u00E2\u0080\u0099 > b suchthat a\u00E2\u0080\u0099, b\u00E2\u0080\u0099 are points of continuity of , and we have V\u00E2\u0080\u0099() \u00E2\u0080\u0098\u00E2\u0080\u0098() V\u00E2\u0080\u0099() (lim (lim cb(z))) V (lim (lim (z)))\u00E2\u0080\u0094y\u00E2\u0080\u0094*-l- z\u00E2\u0080\u0094*y-I- y\u00E2\u0080\u0094x\u00E2\u0080\u0094 z\u00E2\u0080\u0094y\u00E2\u0080\u0094= (urn (z)) v (lirn (z))that is(3.75) (y) llrnq(z) =Hence (3.73) and (3.75) imply (3.72), and the Lemma is completely proved. \u00E2\u0080\u00A2Proposition 3.42 Assume (LFP). Then, the set{ E bv(&) fl {u.s.c.} : (x) = flrn(z),x IR}is a singleton (i = 1,2).Chapter 3. The Free Boundary 72Proof: Assume not and let,qS E bv(b) fl [u.s.c.} such that (x) =j = 1,2, and suppose Yo E IR such that i(yo) 0 be such that i(Yo) + 8 <\u00E2\u0080\u00982(yo) \u00E2\u0080\u0094 , then1Ii(z)&(yo)I <36i, 62 > 0 such that 0 < z\u00E2\u0080\u0094 Yo <6 A 62t I\u00E2\u0080\u00992() \u00E2\u0080\u0094 4(yo)I <8,i.e.0 < z\u00E2\u0080\u0094 Yo < 6 A 62 = 1(z) 0 and 6,62 > 0 such thatz &j(yO).Also, we may assume (for example) that(3.78) ?i(YO) = lim (z),z\u00E2\u0080\u0094*yo\u00E2\u0080\u0094since \u00E2\u0080\u0098z/ satisfies (3.76). Now let y > 0 be such thatlim i(z) > 7 > \u00E2\u0080\u0098j(yo),z\u00E2\u0080\u00944yo\u00E2\u0080\u0094then6> 0 such that z < Yo, z\u00E2\u0080\u0094 oI < = (z) > 7,hence(3.79) inf (z) \u00E2\u0080\u0098y > bj(y0).Ohrnb(Z),and so(3.80) > 0 such that sup \u00E2\u0080\u0098&(z) <7.0< Iz\u00E2\u0080\u0094Yo I Z\u00E2\u0080\u00944Xbut/\u00E2\u0080\u0098jsatisfies (3.76), so there is no loss of generality if we assume (for example)(3.82)\u00E2\u0080\u0098j() = 1im(z).Now let y > 0 be such that 1im\u00E2\u0080\u0099b(z) >7 > \u00E2\u0080\u0098j(), then from (3.82) follows6i>0suchthat0 7 we have(3.84) \u00E2\u0080\u009852 > 0 such that inf 1(z) >.0< IzxI <62Therefore, if 6 = 6 A 62, (3.83) and (3.84) implysup (z) < inf b(z),O(3.87)02 \ O c {(x,,x2) : x2 E (limb(z),bx, ],x, >Z\u00E2\u0080\u0094*21Proof: Let us recall that 0, \0, = R, flA1 (cf. (3.60)). Let P = (x,,x2) e 0, \O,, thenP E R, and hence (ef. (3.35))(3.88) x1 x2), x2 >2(x,).Also, P e A\u00E2\u0080\u0099 and A\u00E2\u0080\u0099 is open, so there is an open ball B(P, r) c A\u00E2\u0080\u0099, but then (cf.Proposition 3.37)(3.89) IB(P,r) fl R, IA\u00E2\u0080\u0099 fl Ru = IOu \ Oil 0i = 0.Claim: x1> lim \u00E2\u0080\u0098i(z).In fact, if not, then x1 < urn b,(z). Thus,Z\u00E2\u0080\u0094\u00E2\u0080\u0099X2(3.90) VE>08>0suchthat inf \u00E2\u0080\u0098i(z)>xi\u00E2\u0080\u0094.O b2(x1) follows(3.91) < r/2 such that sup b2(t) 0 such that 0 < Iz \u00E2\u0080\u0094 x2 <6o ?,bi(z) > x1 \u00E2\u0080\u0094hence (3.92) and (3.93) implyI 1(z) > x1 \u00E2\u0080\u00940 < z \u00E2\u0080\u0094 x2 <6I b(xi \u00E2\u0080\u0094 E) < Z,therefore (cf. (3.35))0< z \u00E2\u0080\u0094x2 <60 = (xi \u00E2\u0080\u0094Eo,z) ER1.Also, since (x1\u00E2\u0080\u0094 80, x2) e B(P, r),> 0 such that 0 < z \u00E2\u0080\u0094 x2 < \u00C3\u00B6 = (xi \u00E2\u0080\u0094 Eo, z) E B(P, r);sofor\u00C3\u00B6=61A60we have(3.94) 0 < z \u00E2\u0080\u0094 x2 <6 = (x1\u00E2\u0080\u0094 60, z) E R1 fl B(P, r),but then, also,{(t,z) E B(P,r):xi\u00E2\u0080\u0094r (r\u00E2\u0080\u00946o)6and this is impossible because of (3.89). So the Claim follows. The Claim and (3.88)prove (3.87). (The proof of (3.87)2 is the same.) \u00E2\u0080\u00A2Chapter 3. The Free Boundary 77Proposition 3.48 Assume (LFF). If,7r(Oj \ 8) where 7r is the orthogonal projection on the x-axis, thenr(O \ ) n (lim = 0fori #j, i,j = 1,2.Proof: Assume not and take i = 1, j = 2 for simplicity. Let P = (i\u00E2\u0080\u0099, ) E Ui \ i withe (i \u00E2\u0080\u0098i(z),i(2)].Z\u00E2\u0080\u0094X2As in the proof of Proposition 3.47 (cf. (3.89)), from P e 0, \ O and 0 \ , = R, fl A\u00E2\u0080\u0099follows that, for some r > 0, B(P, r) C A\u00E2\u0080\u0099 and(3.96) IB(P, r) fl Ru =Thus, we can select \u00E2\u0080\u0098y (, \u00E2\u0080\u0094 r, + r) such that(3.97) 1R(7,.) = 0 a.e. inB7(P,r),whereB7(P,r) = {z: (7,z) E B(P,r)}.In particular, we may fix\u00E2\u0080\u0098yE (, \u00E2\u0080\u0094r,1 +r)fl (urn ,(z),i(2) ,z\u00E2\u0080\u0094+x2and we may assume (for example)(3.98) i(2) = lim ,(z),Z+X2since b, satisfies (3.76) and Proposition 3.42. Then, (3.98) and < 1,(2) imply(3.99)Chapter 3. The Free Boundary 78On the other hand, (3.97) implies(3.100) (-y, z) R, for a.e. z e B7(P, r);therefore it must be(3.101) (7,z) E A\u00E2\u0080\u0099 for a.e. z E B7(P,r),since locally R, is the complement of A\u00E2\u0080\u0099. (In fact, since R, fl (R2 \ Do) = 0 and R1 fl(R0 \ {x = x}) = 0, we can always assume B(P, r) fl (R2 U int(Ro)) = 0, with r smallerif necessary). So from (3.101) we deduce(3.102) > i(z), z > \u00E2\u0080\u00982(7) for a.e. z E B7(P,r).But &, by Lemma 3.44, hence (3.99) and (3.102) imply(3.103) for a.e. z \u00C3\u00A9 (2 \u00E2\u0080\u0094 Thx2)for i = min{r, 6}, and we have a contradiction. \u00E2\u0080\u00A2Corollary 3.49 Assume (LFP). For the non-dense part of R, in A\u00E2\u0080\u0099, c9 \ 0, one has(3.104) 0 \ C {(x1,x2): x1 e (,(x2),x], >1 02 \ 2 c {(x1,x2): x2 (x,),,)],x >Proof: This follows from Proposition 3.47 and Proposition 3.48. \u00E2\u0080\u00A2We can improve Corollary 3.49. In fact, we now show that the inclusions in (3.104)are equalities.Proposition 3.50 Assume (LFP). The non-dense part of R, in A\u00E2\u0080\u0099, oj \ 0, may becharacterized as follows(3.105) Di \ = {(x1,x2) : E(1x2),,(x2)],x2 >( 02 \ \u00C3\u00A3 = {(x1,x2): x2 (ix),],x, >Chapter 3. The Free Boundary 79Proof: Let P= (1,2) E {(x1,x2) : x1 (i(x2),i(], > 2(xi)}, then >1(2), and let us assume that \u00E2\u0080\u00981(2) = urn i(z) (this is possible because of (3.76)).So we havei > \u00E2\u0080\u0098i() = urn i(z) = T1ii i(z) lim i(z) = urn i(z);Z\u00E2\u0080\u0094*X2 Z+X2\u00E2\u0080\u0094therefore, if7 >0 is such that >7> 1(2), then6>0 such that Iz\u00E2\u0080\u0094<6=\u00E2\u0080\u0099l(z)<7 b1 only on a null set imply( \u00E2\u0080\u0094 6,2 + 6): bi(z) > }) = 0,hence from Fubini\u00E2\u0080\u0099s Theorem, (ii) and the definition of R1 followsB(P,r) fl Ri = 0,so that P is in the non-dense part of R1 in A\u00E2\u0080\u0099, 01 \ 0,. Hence (3.105), follows fromCorollary 3.49. (The proof of (3.105)2 is the same.) \u00E2\u0080\u00A2Remark 3.51 We point out that there may be other points Q e Oj such that D(R, Q) =0; these are points where b1 has a cusp and they too have zero Lebesgue density w.r.t.R. The difference between such points and those in O \ 0 is that the latter ones verifya condition even stronger than D(R%, P) = 0, namely (3.89), i.e.(3.106) r > 0 s.t. IR fl B(P, r)I = 0.Chapter 3. The Free Boundary 80We are now ready to characterize the boundary of the new region of action A\u00E2\u0080\u0099, thatis we can finally show point (III).Theorem 3.52 Assume (LFP). The new region of action A\u00E2\u0080\u0099 = int(cl(A\u00E2\u0080\u0099)) is given by(3.107) A1 = {(x,,x2)e : x1 > ,(x2), >with \u00E2\u0080\u0098b, and b2 as in Definition 3.43.Proof: It suffices to recall (cf. (3.64)) that2A\u00E2\u0080\u0099=A\u00E2\u0080\u0099U(aA\u00E2\u0080\u0099\aA\u00E2\u0080\u0099)=A\u00E2\u0080\u0099UUa1\\u00C3\u00A3,1=150 (3.107) follows from (3.33), ?/\u00E2\u0080\u0098j <,= /,, a.e., Proposition 3.50, and the fact that(3.108) {(x,,x2) : 1(x2) x1 > ,(x2),(x ) x2 >2(x,)} = 0(in fact, x1 b,(x2) and x2 2(x,) imply (x,,x2) E R0 = {(x,x)} \u00E2\u0080\u0094 A*, i.e.x e (\u00E2\u0080\u0094oc, x?}, i = 1, 2, but there = \u00E2\u0080\u0098j (cf. Remark 3.46)). \u00E2\u0080\u00A2We can finally show point (IV) as Theorem 3.52 enables us to identify OA\u00E2\u0080\u0099 precisely.In fact, OA\u00E2\u0080\u0099 is essentially obtained by adding all the finite line segments correspondingto the jumps of /\u00E2\u0080\u0098j to the graph of , i = 1, 2, j = 1, 2, j i.[a3 ,ao)Definition 3.53 Let {C}1 and be the points of discontinuity of \u00E2\u0080\u0098, and \u00E2\u0080\u0098b2respectively. Then we set= lim ,(z),(3.109)[i] = iTi ,(z)and similarly\u00E2\u0080\u0098b2[j= 11mb2(z),(3.110)iL2[j] = hni b2(z)=for every j E IN.Chapter 3. The Free Boundary 81Proposition 3.54 Assume (LFP). Then the new free boundary DA\u00E2\u0080\u0099 is given by= 51 U 52= (graph(lO )) U U [i[],i[i1) x {c})(3.111) 31u(graPh(&[O))uU{} x [2[],2[i])).Proof: This is obvious from Theorem 3.52. \u00E2\u0080\u00A2As we saw in Lemma 3.22, all points on the boundary DA\u00E2\u0080\u0099 where a tangent vectorexists belong to the measure-theoretic boundary DMA\u00E2\u0080\u0099. Here we show that this is infact the case for almost every point of DA\u00E2\u0080\u0099. This result is an obvious consequence ofthe rectifiability of the boundaries 0, and 52 and proves point (IV). (Recall that U isrectifiable since R is the subgraph of a function of bounded variation.)Proposition 3.55 Assume (LFP). Then, the topological boundary DA\u00E2\u0080\u0099 and the measure-theoretic boundary DMA\u00E2\u0080\u0099 are the same except for a set of 1-dimensional Hausdorff measure zero, i.e.(3.112) H\u00E2\u0080\u0099(DA\u00E2\u0080\u0099 \ DMA\u00E2\u0080\u0099) = 0.Proof: It suffices to show that there exists a definite tangent to M\u00E2\u0080\u0099 almost everywherewith respect to the 1-dimensional Hausdorif measure H\u00E2\u0080\u0099 in 112 (cf. Lemma 3.22). Butand \u00E2\u0080\u0098L\u00E2\u0080\u00992 are functions of locally bounded variation (in the usual sense), hence 0, and 52are locally rectifiable curves, and the measure H\u00E2\u0080\u0099 coincides with the arc-length s. Also,a result due to Tonelli (cf. [Tn]) guarantees that the classical formula(s\u00E2\u0080\u0099(t))2 = (x\u00E2\u0080\u0099(t))2 + (y\u00E2\u0080\u0099(t))2is valid for every rectifiable curve and a.e. with respect to the parameter t which isarbitrary (if, locally, x = x(t), y = y(t) is a parametric representation of the curve). Inparticular, if we choose the arc-length s as parameter, we obtain(x\u00E2\u0080\u0099(s))2 + (y\u00E2\u0080\u0099(s))2 = 1 s-a.e.Chapter 3. The Free Boundary 82and hence x\u00E2\u0080\u0099(s), y\u00E2\u0080\u0099(s) exist a.e. assuring the existence of a definite tangent almosteverywhere with respect to s.Finally, from Proposition 3.55 and Remarks 3.25 and 3.26 we obtain the regularity ofthe entire boundary of the new region of inaction, i.e. we show the validity of point (V).Theorem 3.56 Assume (LFP). Then the new region of inaction Al is of locally finiteperimeter and its boundary DA\u00E2\u0080\u0099 is countably 1-rectifiable, i.e.(3.113) 0)11 cUMUNwhere H\u00E2\u0080\u0099(N) = 0 and each M1 is a 1-dimensional embedded C1 submanifold of 1R2. \u00E2\u0080\u00A23.5 Finite Perimeter of A\u00E2\u0080\u0099: a Verification of (LFP)In the previous two sections we obtained the regularity of the free boundary arising inthe control problem defined by (2.1) under the assumption (LFP); that is, we assumedthe region of inaction A\u00E2\u0080\u0099 to be of locally finite perimeter. In order to complete thischapter we must show that such an assumption is, after all, reasonable and verifiable.We shall restrict ourselves to the case where the diffusion matrix a is nondegenerate.Such a condition naturally implies the coercivity (see below) of the bilinear form a(u, v)associated with the operator Au of (2.40), and this allows us to show (LFP) by means ofa localization of a result obtained by Brezis and Kinderlehrer [BK] in the framework ofvariational inequalities with obstacles relative to locally coercive vector fields.In addition to the assumptions stated in Section 2.1 and Section 2.2, we now assumethe following(3.114) uo is positive definite;(3.115) f e C2(1R)Chapter 3. The Free Boundary 83(3.116) f and V(f) never vanish simultaneously (i = 1,2).(It should be noticed that we already had f e C\u00E2\u0080\u009D1(1R2)as this follows from the growthconditions (2.4)\u00E2\u0080\u0094(2.6) by using the same arguments as in Proposition 2.2 and Theorem 2.3.) Let W\u00E2\u0080\u00992() be the closure of C\u00E2\u0080\u0099\u00C2\u00B0(f) in W\u00E2\u0080\u009D2(), for any open set c 1R2.Definition 3.57 [[Fr2], p.15] A bilinear form a(u, v) is said to be coercive on W\u00E2\u0080\u00992(f) if(3.117) 3v>O such that a(u,u) vIIuIIi,2 for every u E W\u00E2\u0080\u00992(),where II 111,2 is the norm in 1ITvl2()In particular, we will consider the bilinear form a(u, v) associated with the operatorAu, i.e.(3.118) a(u, v) := J { (uu*)jjuxvx \u00E2\u0080\u0094 gjuv + dxi=1for u, v e W1\u00E2\u0080\u00992(fZ), with fZ open in R2 (to be chosen later). Let us recall a few knownresults. We start with an obvious lemma.Lemma 3.58 [[WCM], Lemma 4.3]. Assume (3.114). Let 2 c 1R2 be an open ball andlet a(u, v) be defined by (3.118). Then, a(u, v) is coercive on W\u00E2\u0080\u009D2().Proof: This follows from p> 0, ou positive definite and f uua dx = 0 for u E W\u00E2\u0080\u009D2(!),i=1,2. \u00E2\u0080\u00A2Definition 3.59 Let a be as in (3.118) and let(3.119) 1K(1) := {v e W\u00E2\u0080\u009D2() : v 0 a.e. in }.We say that w is a local solution of the variational inequality(3.120) a(w,v\u00E2\u0080\u0094 w) (f,v \u00E2\u0080\u0094 w) Vv E 1K(Z),Chapter 3. The Free Boundary 84if wE 1K() and we have(3.121) a(w, ij(v \u00E2\u0080\u0094 w)) f fij(v \u00E2\u0080\u0094 w) dx Vv E I((), e C(), 0.Theorem 3.60 [[WCM], Theorem 4.5)]. Assume (3.114) and (3.115). Let be an openball such that cl() C S where(3.122) S := {x e ]R2 : > 0,j $ i},then is a local solution of (3.120), i = 1,2.Proof: (sketch) This follows from the fact that i% can be approximated by the uniquesolution ye eC2!L(1R), for every i e (0, 1), of the penalized problem(3.123) Av+ = f a.e. in R2,E i=1where the uniqueness is accomplished among all continuous functions of at most polynomial growth. (Notice that (3.123) is similar to our (2.44) but with the smooth penaltyfunction j5\u00E2\u0080\u0099 replacing the non-smooth (.) of (2.44). This device yields more regularityfor the solution.) Then, the Schauder theory for classical solutions (which applies thanksto (3.114) and (3.115)) shows that yC e C3I4(1R2); hence (3.123) can be locally differentiated, and in the limit as \u00E2\u0080\u0094* 0 one obtains (3.120). (For the detailed proof see thereference above; there it is assumed f \u00E2\u0082\u00AC C3, but, in fact, f E C2 is enough.) iTheorem 3.61 [[1+2], problem 5, p.30 and problem 1, p.44; [WCM], Theorem 4.6] Assume (3.114) and (3.115). Let 2 be an open ball such that c1() C S, then(i) il e W2\u00E2\u0080\u0099();(ii) A1 f, \u00E2\u0080\u0098Ii 0, (Ai\u00E2\u0080\u0094f)ii = 0 a.e. in .Chapter 3. The Free Boundary 85Proof: (sketch) (This result is proved in the second reference mentioned above.) Letbe an open ball such that cl() C Q C cl(O) C S. The fact that \u00C3\u00B12, is a local solution of(3.120) implies \u00C3\u00B1r e W1\u00E2\u0080\u0099() for every 1

0). \u00E2\u0080\u00A2Remark 3.67 As a matter of fact, it is possible to show that On = 0 under the conditions (3.114), (3.115) and (3.116) (cf. [Fr2], Theorem 3.4 and Theorem 3.5, p.155).We are now ready to prove (LFP).Chapter 3. The Free Boundary 91Theorem 3.68 Assume (3.114), (3.115) and (3.116). Then the region of inaction A\u00E2\u0080\u0099is of locally finite perimeter, i.e. (LFP) is verified.Proof: Since f E C(1R2) and f3\u00E2\u0080\u0099r <0, we can cover Or with open balls such that(3.140) holds there, i.e.hr \u00E2\u0080\u009411RrflfZ = . a.e. in ftJ XThen, 11RV E BVi0(), r = 1,2, (by Theorem 3.64, (3.115) and (3.139)); also OA\u00E2\u0080\u0099 =O U 02, hence the assertion of the Theorem follows. \u00E2\u0080\u00A2Clearly all the results of Sections 3.3 and 3.4 hold in the present setting (since (LFP)holds for A\u00E2\u0080\u0099). However, under the assumptions (3.114)\u00E2\u0080\u0094(3.116), Proposition 3.54, canbe considerably improved thanks to the greater regularity of ftr as we show in the nextsection.3.6 Higher Regularity of the BoundaryIn this section we assume (3.114)\u00E2\u0080\u0094(3.116) and we show (with the notation of the previoussections) that the function which defines the boundary 0 of the region of inactionR, is Lipschitz continuous, and therefore = The proof is a generalization of aresult concerning the regularity of the free boundary of a filtration problem (cf. [Fr2],Theorem 6.1, p.l7\u00E2\u0080\u00997). The arguments of the proof are based on PDE methods (and forthese we need to assume (3.114), (3.115) and (3.116)), and also on the geometry ofthe problem (that is, on the results of Section 3.2). As all the results of Section 3.2can be easily extended to the n-dimensional case, the regularity theorems which we aregoing to obtain hold for the n-dimensional monotone follower problem too. However, forsimplicity and consistency with the rest of this thesis, all the statements and the proofsbelow will be limited to the 2-dimensional case.Chapter 3. The Free Boundary 92We will need the following lemma which provides us with some basic properties ofLet us recall that\u00E2\u0080\u0098\u00C3\u00BCXr e C\u00E2\u0080\u009D(Sr) (cf. (3.124)).Lemma 3.69 Assume (3.114) and (3.115). Let ! be an open ball in Sr which intersectsOr\O\u00C3\u00BC, letP e A\u00E2\u0080\u0099fll be such thatdist(P,\u00C3\u00B4r\Oo) <6, dist(P,O) o >0, letM> 0be such thatI1)jj(r)I M in ,for all i, j. Then(3.145) iixr(P) M6 Gb2,(3.146) Vu(P)I C6,where C = Cfr0, IU2rIICI1()), r = 1, 2.Proof: The proof is essentially that of Lemma 3.2 and Corollary 3.3 in [Fr2], p.155.Recall that= 0, V\u00C3\u00B1 = 0 on \u00C3\u00B6r \ 0o(by Lemma 3.66). Then (3.145) follows from the Taylor formula (assume 6 < o)(P\u00E2\u0080\u0094Q).e r0 r(P) = xr(Q) + Dj(r)(Q)(P\u00E2\u0080\u0094 Q) e + f j Di()(Q +te)dtdr,with Q E Or \ O, e is the unit vector in the direction of QP, and D2 the directionalderivative in the direction e.Now another application of the Taylor formula yields0 r(P + /Ai6\u00E2\u0080\u0099e) = ftr(P) + Dj(i%xr)(P)/7i6\u00E2\u0080\u0099T+j j D(ui)(P+te)dtdTwith e3 in the direction for which D(\u00C3\u00B1j(P) = IVir(P)I and 6\u00E2\u0080\u0099 < 6 such that/M6\u00E2\u0080\u0099 <. Hence0 Mb2\u00E2\u0080\u0094IV\u00C3\u00BCr(P)I6/:+ M2(6\u00E2\u0080\u0099)Chapter 3. The Free Boundary 93i.e.Ixr (p)I 6(1 + M)/Ai/6\u00E2\u0080\u0099,i.e. (3.146) follows.Theorem 3.70 Assume (3.114), f C3(1R2) and (3.116). Then the functions \u00E2\u0080\u0098b, andare continuous everywhere and locally Lipschitz away from the corner point a0. Inparticular, the free boundary DA\u00E2\u0080\u0099 is given by(3.147) DA\u00E2\u0080\u0099 = 01 U 02 = (graph(,[0 ))) u (graph(2[0))).Proof: (Compare with [Fr2], Theorem 6.1, p.177.) Fix i = 1 for simplicity and recallthatE C\u00E2\u0080\u009D(c), i = 1,2,for every open ball C A\u00E2\u0080\u0099 (cf. (3.124)); hence(3.148) n e C\u00E2\u0080\u009D(Q),and so also(3.149)Then, from (3.148) and f E C2\u00E2\u0080\u009D() follows(3.150) E C4a(c) for every open ball 2 C A1,by elliptic regularity (cf (3.114) and [GT], Proposition 6.17, p.109). This enables us todifferentiate the Bellman equation once more; i.e. we get\u00C3\u00A7 Al\u00E2\u0080\u0094inand f 0 (by convexity) implies(3.151) Aii1, 0 in A\u00E2\u0080\u0099.Chapter 3. The Free Boundary 94Also, 0 (again by convexity) and = 0 on 0, \ 0o (by (3.143)). Now let bean open ball in S, = R, U A\u00E2\u0080\u0099 such that fl D 0, then we apply the strong maximumprinciple (cf. [GT], Theorem 3.5, p.35) toAit1, O infflA1,=O onflD1,0 in Q,E C2a(! n A\u00E2\u0080\u0099) n C\u00C2\u00B0\u00E2\u0080\u009D(cz),and we conclude (p> 0 is used here)(3.152) > 0 in n A\u00E2\u0080\u0099,since if not, then the minimum (which is zero) would be achieved inside ! and thiscontradicts the maximum principle.We recall that f < 0 on D \ Do (by Theorem 3.64); hence, by continuity, f, <0 ina neighbourhood of any point of O \ 0o, and we assume(3.153) f, <0 in ftLet Q = (x?, x) E 9 fl and let R> 0 be such that there exists b> 0 for whichDR := {(x,,x2)E A\u00E2\u0080\u0099 : 1x2\u00E2\u0080\u009441 0, F> 0, and IHI 1 (with the constants K and F to be chosen later). Thenwe haveAw = Kf11 + Hf,2 \u00E2\u0080\u0094 Ff1,Chapter 3. The Free Boundary 95and from this follows, for F sufficiently large (independently of K and H),(3.154) Aw > > 0 in D2R,since f11, fria are bounded in l and (3.153) holds. Also(3.155) w=0 onODRflO,,and (3.152) implies(3.156) w > 0 on ODRI fl {dist(.,Oi) > 6}, for all R < R\u00E2\u0080\u0099 2R,if K K(6) with 0 < 6 < 1, a small number to be determined later. If we show(3.157) w> 0 on ODR fl {dist(., 0,) 6} fl A\u00E2\u0080\u0099,then by applying the maximum principle to (3.154)\u00E2\u0080\u0094(3.157) we will conclude(3.158) w > 0 in DR.We show (3.157) by contradiction. Assume (3.157) is false and let P E OD fl A\u00E2\u0080\u0099 besuch that(3.159) w(P) <0 and dist(P,O,) 6.Now set\u00E2\u0080\u0098th(P) = w(P) + vP\u00E2\u0080\u0094 P2, P E cl(D2R),with v> 0 small enough to guaranteeA(vIP \u00E2\u0080\u0094 I2) > \u00E2\u0080\u0094EforE asin(3.154) (thiscanbedonebecauseof(3.114), i.e. ER2with > 0). Thus,(3.160) A3 > 0 in D2R,(3.161) 0 onOD2RflO,,Chapter 3. The Free Boundary 96and(3.162) Z\u00E2\u0080\u0099 > 0 on 9D2R fl {dist(., a) > 6}as this follows from (3.156). Therefore, if we show(3.163) > 0 on OD2R fl {dist(., 0,) 6} fl A\u00E2\u0080\u0099,then by aplying the strong maximum principle to (3.160)\u00E2\u0080\u0094(3.163) we will get>0 inD,which contradictsand hence we will have proved (3.157). So let us show that (3.163) holds. Let P EOD2R fl {dist(., a,) 5} fl A\u00E2\u0080\u0099, and recall thatC\u00E2\u0080\u009D(Q), = 0 on 0, Vi1 = 0 on 0 \ a,by (3.143) and (3.124); then, by Lemma 3.69 followsf&,1(P) C62 and IV1(P)I C6.Hence1Y(P) = Ki21(P) + Hf12(P) \u00E2\u0080\u0094 FI1X1(P) + viP \u00E2\u0080\u0094 P12Ki11(P) \u00E2\u0080\u0094 HC6 \u00E2\u0080\u0094 FC62 + viP \u00E2\u0080\u0094 Fl2> K11X(P) \u00E2\u0080\u0094 C5(1 + F) + yR2yR2since 6 < 1 and l \u00E2\u0080\u0094 1I R. Then, certainly z(P) > 0 if we choose 6 0 inDR,Chapter 3. The Free Boundary 97i.e.K11X + Hii12 \u00E2\u0080\u0094 FI1X1 > 0 in D;but > 0 in A\u00E2\u0080\u0099, hence we have(3.164) KILX1X, + Hi12 > 0 in DR,which means that i% increases along lines of slope H/K in DR, for any H such thatI Hj 1. However, \u00C3\u00B1 = 0 on 0, therefore there exists a cone 7+(Q) with vertex Q, andangle 2/3 = 2 arctan(), and with axis parallel to thex1-axis such that7(Q) fl DR C A\u00E2\u0080\u0099.The same holds for any P e 0, fl VQ, VQ being a small neighbourhood of Q, i.e.(3.165) \u00E2\u0080\u0098y(P) fl VQ C A\u00E2\u0080\u0099, VP E VQ fl 0.But we do know that R, is a portion of the subgraph of /\u00E2\u0080\u0098,(x2), hence it must also be(3.166) y(P) fl VQ C R,, VP E Vc fl 0,,if:= {(x,,x2): (\u00E2\u0080\u0094x1,x2)eThus is Lipschitz continuous inr2(VQ) (with r2 being the projection onto thex2-axis);in fact,(3.167) I1(x2) - &1(y2)I max{tan/3,tan(- /3)} = max{1/K,K}for every .x2, Y2 1r2(VQ). \u00E2\u0080\u00A2Finally, from the Lipschitz continuity follows even greater regularity by the applicationof classical regularity results (compare with [Fr2], Theorem 6.2, p.179).Theorem 3.71 Assume (3.114), f e C3(1R2), and (3.116). Then, for i = 1, 2, jChapter 3. The Free Boundary 98(i) E C\u00E2\u0080\u0099((x?,+oo)) and\u00C3\u00B1 e C2(A\u00E2\u0080\u0099 u(0\0o));(ii) f, Cm,a({f, 0 such that f12 + M1f22 0 in 1R2(M2 > 0 such that f12 + M2f11 0 in 1R2Williams, Chow and Menaldi proved that the points in > 0} have positiveLebesgue density with respect to the coincidence set = 0}, and hence obtained theregularity by applying Caffarelli\u00E2\u0080\u0099s result. However, our Section 3.6 shows that it sufficesto assume (3.114), f e C3(1R2), and (3.116) in order to obtain a value function i soregular that its corresponding free boundary is, in fact, of class C2,. Again, this resultgeneralizes to the n-dimensional case (as does Section 3.2).In conclusion, there is essentially no problem once we assume (3.114), f e C3(1R2), and(3.116) (except perhaps at the corner point Oo); but under the more general hypothesesof Sections 2.1 and 2.2 the free boundary could be really \u00E2\u0080\u009Cnasty\u00E2\u0080\u009D, and hence difficult tostudy (PDE methods no longer being applicable because of degeneracy and less regularityof the cost rate f). That is why (LFP) seems to be the right way to study 0A1 andanalyze its \u00E2\u0080\u009Cbad\u00E2\u0080\u009D points (essentially those points which are not in DMA1). We believethat (LFP) is a property intrinsic to the kind of singular control problem studied in thisthesis (perhaps a consequence of the convexity and growth conditions), and although ourverification of (LFP) assumes (3.114)\u00E2\u0080\u0094(3.116) (cf. Section 3.5), we feel that it should beChapter 3. The Free Boundary 101possible to show (LFP) under the more general conditions of Sections 2.1 and 2.2. Weleave this for future work.Chapter 4The Optimal Control4.1 IntroductionThroughout this Chapter we assume (LFP). In Section 4.2 we deduce, from the geometryof the new region of inaction Al, the properties that a control k would be required topossess in order to be optimal. In Section 4.3 we construct a \u00E2\u0080\u009Cpotentially optimal\u00E2\u0080\u009D controlk by making an essential use of the characterization of A\u00E2\u0080\u0099 in terms of the functions /\u00E2\u0080\u0098(cf. (3.107)). ic is obtained as the solution of a fixed point problem; the existence of thissolution is guaranteed if one assumes that the condition (LIP) below holds in a smallneighbourhood of the corner point Po (i.e., the only point of intersection of graph(b,)and graph(\u00E2\u0080\u00992), and also the only point of cl(A\u00E2\u0080\u0099) where V = 0). In particular, k is alsoadapted and uniquely determined if (LIP) holds true. Then, in Section 4.4 we assume(LIP) and we show that k is the unique optimal control of the original problem (2.3);this is done by means of a generalization of Krylov\u00E2\u0080\u0099s proof of Ito\u00E2\u0080\u0099s formula for Sobolevfunctions. Finally in Section 4.5 we give conditions under which (LIP) holds.4.2 The Optimal Control: a Heuristic DescriptionWe recall that in Chapter 3 it was shown that the region of inaction A\u00E2\u0080\u0099 can be redefined(cf. (3.57)) asA\u00E2\u0080\u0099 = int(cl(A\u00E2\u0080\u0099)),102Chapter 4. The Optimal Control 103so that the dynamic programming equation holds a.e. in A\u00E2\u0080\u0099 (cf. (3.67)), i.e.= f a.e. in A\u00E2\u0080\u0099.Moreover,OA\u00E2\u0080\u0099=\u00C3\u00A3,u\u00C3\u00A32, \u00C3\u00A3,n\u00C3\u00A32=00withO c OR,(cf. (3.57)\u00E2\u0080\u0094(3.59) and Lemma 3.38), hence from Theorem 3.11,(iv) followsonO,\Oo,(4.1) V 0 onO0,on 02 \ Do,if e, := (1,0) and \u00C3\u00AB2 := (0, 1). Finally we recall that (cf. Proposition 3.54)= graph( q+) u [i[},i[r]) Xfoij, i,j=1,2,and(r ifil,TEN.1r ifi=2Therefore, since 0, C OR2, one has (by definition (3.3) of \u00E2\u0080\u0098/\u00E2\u0080\u0098j and Definition 3.43 of \u00E2\u0080\u0098/\u00E2\u0080\u0098j,i = 1,2)1(x1,X2) = \u00C3\u00BC(i[r],(r) V(x,,x2)e [,[r],,[r]) X(4.2)\u00E2\u0080\u0098ii(xj,x2)= ir,\u00E2\u0080\u00992[rJ) V(xi,x2)E {.} x [&2[L],\u00E2\u0080\u00992[r]),for every r E IN.All the information so far collected suggests that optimality could be achieved in(2.3) by a control k that keeps the controlled process within cl(A\u00E2\u0080\u0099) after time zero; (ifXo e (Al)c, an instantaneous jump of the controlled process to a suitable point of theboundary would be required). More precisely a \u00E2\u0080\u009Cpotentially optimal\u00E2\u0080\u009D control couldbe a control k := (k, k?) E V such thatChapter 4. The Optimal Control 104(i) k =[2(xg) \u00E2\u0080\u0094 X];(ii) k increases only at the boundary \u00C3\u00B4j with dk\u00C2\u00B1V(X) for all t such that X E O;\u00E2\u0080\u0094- if ii e [i[t],i[r]) and X?_ 13r,(iii) Lk = for t > 0;0 otherwise,(iv) E {f\u00C2\u00B0\u00C2\u00B0 f(t)e_Ptdt} <00,where zk = k \u00E2\u0080\u0094 k_, r IN, {i,j} = {1, 2} and JC = (\u00E2\u0080\u0098, -) is the controlledprocess starting from x E JR2 at time t = 0 and corresponding to the control k. In fact,heuristically, if Ii were C2 then by applying the Ito formula for semimartingales (cf. [My],p. 278) to (t)ePt we would obtain(x) = E{j Aii(X3)e8ds+(4.3)\u00E2\u0080\u0094 j eVui(X8) (\u00C3\u00ABdk +\u00C3\u00AB2dk)- e [i3)- \u00C3\u00BC() - Vi3). (\u00C3\u00ABiLk +\u00C3\u00AB2Lk)] }.O 0, k were defined to be the smallest positive number suchthat the entire path of X3 + kej up to time t is contained in cl(((Ro U E) \u00E2\u0080\u0094 Ke)c), fori,j such that {i,j} = {1,2}. We set\u00E2\u0080\u00A2 X:=X+k\u00E2\u0080\u0099\u00C3\u00ABi+k\u00C3\u00AB2 t>0,Chapter 4. The Optimal Control 107and we proceed to give an explicit representation of k. We introduce the followingstopping time(4.6) T\u00E2\u0080\u0099 := T\u00E2\u0080\u0099 A T,\u00E2\u0080\u0099 A T,where, for i = 1,2,1inf{t>0:Xt(w)Ei\Oo}, ifPEint(A\u00E2\u0080\u0099U),ji,(4.7) T (Lb\u00E2\u0080\u0099) :=0, if PER0UR;(4.8) T(w) inf{t >0: X(w) e o}, if P R0,10, ifPER0;with the usual convention that inf 0 = +oc.It is clear that k represents the least intensity of the pushing in the direction \u00C3\u00ABneeded in order to keep the process X inside cl(A\u00E2\u0080\u0099). Hence, at least up to time r, bothk and k? are well defined; in fact,k\u00E2\u0080\u0099 =0=k Vt t T, E IN,\u00E2\u0080\u0094=[2 (x + \u00E2\u0080\u0094 (x + k2)] +with10 ifP\u00C3\u00A9A1,k1_ = (k0 ifP\u00E2\u0080\u0099A\u00E2\u0080\u0099.To see this let = r (for example), then J( i for all t < T72, hence(x? + < X\u00E2\u0080\u0099 + k for all r\u00E2\u0080\u0099 t < r\u00E2\u0080\u00991 and k = k. Thus, (4.18) holds, i.e.\u00E2\u0080\u0094= { (x + \u00E2\u0080\u0094 (x + = 0 v r t T?1.Chapter 4. The Optimal Control 109On the other hand,. since r\u00E2\u0080\u0099\u00E2\u0080\u0094 \u00E2\u0080\u0094 = 2 (x + \u00E2\u0080\u0094 (x + \u00E2\u0080\u0094)=2 [ r] - (x + if + k E {} x [2 [LI , [r I)( 0 otherwise;hence the process X + k is kept inside cl(A\u00E2\u0080\u0099) if satisfiesmax VT 0 is any time. Notice that in (4.19) we have k instead of k_; in fact, k = 14since b1 and are continuous at zero (cf. Lemma 3.9 and the definition of \u00C3\u00A7b). Then,the problem reduces to searching for a solution of= maxO0 and for i = 1,2,(LIP) : \u00E2\u0080\u0098/\u00E2\u0080\u0098j is Lipschitz continuous in [\u00E2\u0080\u0094a, a] with Lipschitz constant \u00C2\u00A3ip(1)< 1;(NIN) /\u00E2\u0080\u0098j is non-increasing and continuous in [\u00E2\u0080\u0094a, a].Note that the non-increasing property of is the condition supposedly established in[MRb].Recall that, for simplicity, we are assuming Oo = {Po} = {(0, 0)}. Of course, both(LIP) and (NIN) are always satisfied in [\u00E2\u0080\u0094a, 0], since \u00E2\u0080\u0098/\u00E2\u0080\u0098j is constant there (cf. Lemma 3.8)and hence jj = in [\u00E2\u0080\u0094a, 0].From now on we will always assume that either (LIP) or (NIN) holds true for Let1(x2)= maxi(y),2(x1)= max2(y),yx1Chapter 4. The Optimal Control 111then graph(coll(0))lies above graph(2l(O)) and both graphs lie within the firstquadrant. Moreover,A := {(x1,x2)E 0o + A* : x1 >1(x2), > Y2(Xl)} 0,as this follows from the definition of cpj and (3.86). The characterization of Al given by(3.107) implies A C A\u00E2\u0080\u0099. Now we choose\u00E2\u0080\u00A2=(1,f2)e A\u00C2\u00B0 such that\u00E2\u0080\u00A2 R> 0 such that P\u00C2\u00B1R\u00C3\u00AB, \u00C2\u00B1R\u00C3\u00AB2 E Afl[(O,c) x (0,)j;notice that this choice of P and R impliesfor {i,j} = {1,2}. Finally we set\u00E2\u0080\u00A2 L31= { (x1,2) : lxii O:Yt(w)l3R}, wEe;\u00E2\u0080\u00A2 mh(6) := sup jh(t\u00E2\u0080\u0099)\u00E2\u0080\u0094 h(t\u00E2\u0080\u009D)i for h e C[0,T] and 6 > 0;It\u00E2\u0080\u0099\u00E2\u0080\u0094t\u00E2\u0080\u009DIb-- ( 12) if t 0) where(4.22) ffi(6) := c + /6log(1/6)],where c is some constant. Let 6 > 0 be such that 6, . 0 as n \u00E2\u0080\u0094* cc, and set\u00E2\u0080\u00A2 r(w) := inf{t > 0: sup [mYI[0 ()(6) \u00E2\u0080\u0094 1h(6)] o}, for n E IN;\u00E2\u0080\u00A2 r(w) := TR(W) A r(w), for n E IN, w ENotice that r is a stopping time. Also, T TR A T a.s. because of P. Levy\u00E2\u0080\u0099s result. Let\u00E2\u0080\u00A2 i(n,w) := the process w) stopped at r;f m,. (2th(6)) + ii(6) if (NIN) holds for j,\u00E2\u0080\u00A2\u00E2\u0080\u00982(6):=\u00C3\u00A7I 2Cip(\u00E2\u0080\u0099i/\u00E2\u0080\u0099)th(6) + th(6) if (LIP) holds for \u00E2\u0080\u0098j,1 0 if (NIN) holds for j,:= \u00C2\u00A3ip()th(6) \u00C2\u00B1 th(6) if (LIP) holds for j,1 \u00E2\u0080\u0094for i=1,2,and6>O;\u00E2\u0080\u00A2 C(6) := max{Jj(6),.j(6) : i = 1,2} for 6> 0;\u00E2\u0080\u00A2 N : {(h1,h2) E (C[0,t])2: (i) h 0,14 = 0;(ii) h is non-decreasing;(iii) mhl(6) C(6);(iv) (h,h) E t3p Vs E [0,t];for i,j = 1,2 and for all 6 Efor n e IN, t> 0, and write N for N\u00E2\u0080\u0099 where T> 0 is fixed (but arbitrary);\u00E2\u0080\u00A2 ir(h\u00E2\u0080\u0099,h2):= h for (h\u00E2\u0080\u0099,h2) E (C[0,T])2 and i = 1,2;Chapter 4. The Optimal Control 113\u00E2\u0080\u00A2 I c is such that Y(w) is continuous for w 0 I.Proposition 4.1 Let (C[O, T])2 be endowed with the product topology induced by thesup-norm topology of C[O, T]. Then for each n E IN, N is a compact convex subset of(C[o, T])2.Proof: The set r(N) of elements of C[O, T] is equicontinuous by the definition ofN since C(6) < C(6) for 6 6,. Also, r(N) is bounded, as follows from (iv) of thedefinition of N.It remains only to check that ir(N) is closed in C[O,T]. Let hr E 7r(N) andh e C[O, T] be such that hr \u00E2\u0080\u0094* h in the sup-norm. Then, clearly h satisfies (i) and (ii) ofthe definition of N. Also, (iii) holds for h since mhr(\u00C3\u00B6) < C(6) for every 0 < 6 < 6, andr E IN, and since m.(t5) is a continuous function in C[0, Tj. Finally, (iv) follows triviallyfrom hr \u00E2\u0080\u0094 h as r \u00E2\u0080\u0094> cc. Hence h e ir (Na). Thus ir(N) is compact in C[0, T], j = 1,2.Hence N is compact in (C[0,Tj)2,since N = ir1(N) x ir2(N).The convexity of N is clear from the definition of N. \u00E2\u0080\u00A2Definition 4.2 For every n E IN, h e C[0, T] and w E we set(4.23) T(,t)(h) := max [ (2() + h8) \u00E2\u0080\u0094O?2(n)+hthen, if (NIN) holds, we have(4.32) (i(n) + h1) \u00E2\u0080\u0098i (2(n) + h)since p2(n) + h E [\u00E2\u0080\u0094,] for every r E [O,Tj (by Remark 4.3). Hence from (4.29) and(4.32) follows(4.33) 0 (T(h)) \u00E2\u0080\u0094 (T,(h2)) m1(,)(6) < (6),and again we obtain (iii). On the other hand, if (LIP) holds, then (4.29) implies0 (T(h2)) \u00E2\u0080\u0094 (T(h2))(4.34) \u00C2\u00A3ip(i) [I(n) \u00E2\u0080\u0094 72(fl)I + Ih\u00E2\u0080\u0094 hIJ + i(n)\u00E2\u0080\u0094\u00C2\u00A3ip(l)m2()(6) + \u00C2\u00A3ip(11)mh(6) + ffl1()(6),with mh(6) C(6). So, ifmh(6) <(6)[nw) +1\u00E2\u0080\u0094\u00C2\u00A3ZP@i)then from (4.34) followsmTi(h2)(b) (6).If(4.35) (6) 1(y2)_y1Hence(1+ , + e (\u00C3\u00B1i Ai) fl (ff2 U Al) = cl(A1)and again we obtain (4.58) from (4.54) and E Sp, a.s..Now we can finally answer the crucial question posed at the beginning of this section;that is, \u00E2\u0080\u009Chow should k\u00E2\u0080\u0099 and k2 be defined after J reaches the corner point (0,0) ?\u00E2\u0080\u009C. Thefixed point of t (T is defined by the RHS of (4.55)) is actually a function of \u00E2\u0080\u0098(w), i.e.of Y(w) or of Y e )) where(4.62) 3) { e (C[0,T])2 : Y(0) = o},therefore we write it as \u00E2\u0080\u0098(Y); that is,(4.63) Y)=+ \u00E2\u0080\u0098(Y)) \u00E2\u0080\u0094Note that if r is a stopping time such that= X(w) + (w) = (0,0) =thenXs+r(W) + k(w) =X3+(w) \u00E2\u0080\u0094 X(w) += X3+() \u00E2\u0080\u0094 Xr(W) =: YT(w) E 3).Hence we set(4.64) t(T,W) :=Chapter 4. The Optimal Control 126For w E {r\u00E2\u0080\u0099 = r01} let(4.65) 0\u00E2\u0080\u009D\u00C2\u00B0(\u00C3\u00BC) TA inf{t > 0 : YtTl(,) B}(4.66) := r\u00E2\u0080\u0099(w) + 0\u00E2\u0080\u009D\u00C2\u00B0(w).For t E [T\u00E2\u0080\u0099(w),r1\u00E2\u0080\u009D(w)j we define(4.67) kt(w) := \u00E2\u0080\u0098(\u00E2\u0080\u0098) +5\u00E2\u0080\u0099j_7i(T1,w).Notice that \u00E2\u0080\u0098\u00E2\u0080\u0098 = on [0, O101 and \u00E2\u0080\u00985\u00E2\u0080\u0099s is continuous by construction. Then, ifXTI,1(w) (0,0) we set(4.68) T2(W) :=otherwise we set(4.69) O\u00E2\u0080\u009D(w) := TA inf{t > 0 : YT\u00E2\u0080\u0099I(w)(4.70) r\u00E2\u0080\u009D2(w) := r\u00E2\u0080\u009D(w) + O\u00E2\u0080\u009D(w),and fort E [\u00E2\u0080\u0098r\u00E2\u0080\u009D(w),r\u00E2\u0080\u009D2(w ] we define(4.71) j) := ki,i(i) + _i,i(r\u00E2\u0080\u009D,w),etc. It is clear that if (4.68) holds, then X2() cl(A\u00E2\u0080\u0099) \ 0o as this follows fromProposition 4.20. Hence we set2fr() if X2(w) E \u00C3\u00B11 \ 0o, i = 1,2,(4.72) -r (w) =:L. T(W) if X2(w) E A1,as well as(4.73) k72(w) :=since - is continuous and X2_ E (\u00E2\u0080\u0094o, o.Chapter 4. The Optimal Control 127Now define k inductively. Assume we have defined T\u00E2\u0080\u0099 and k on [0, Ta]. Then:=f w {r = T}It E(4.74) whereT1() := r(w) A rr1(w) Arr1(),T;+\u00E2\u0080\u0099(W) := inf{t T\u00E2\u0080\u0099(W) : X(w) +k(w) E E}, p = 1,2,0,E1:=Ei\Oo, B2:=R\Oo, B0:=8.(Notice that T\u00E2\u0080\u00992 = T\u00E2\u0080\u0099 implies = Tn.)kt(w) :=+ w) + )}fwe{T=T}if(t e [T(W),T+\u00E2\u0080\u0099(W))where:=(4.75) Oni(,) T A inf{t > 0 : YT(w):= T\u00E2\u0080\u0099(w) +O\u00E2\u0080\u0099(w) = n(W) +\u00E2\u0080\u0098(w),l(w) := min{l E IN : k,1(w) (0,0)},I T\u00E2\u0080\u009D1\u00E2\u0080\u009D(W) if Ern,in (w):= for p = 1, 2,3,(+00 otherwise,Tn+(W) := +i(w) A T+l(w) A 72+l(w),i:=ii\Oo, B2:=R\Oo, 3:=A1.Chapter 4. The Optimal Control 128(When (4.75) holds, kn+1() = i+i_(w) since \u00E2\u0080\u00985\u00E2\u0080\u0099 is continuous as < cr.)Ic(w) :=Iii \u00E2\u0080\u0098 ._ i.intjW)\u00E2\u0080\u009CT(w)\u00E2\u0080\u0094 W+ max (X(w) + fl()(w)) - (X(w) +1we{T=r}, j=1,2if(4.76) It e [r(oJ),T(w))where:= \u00E2\u0080\u0098(w) AT41(w), i j , i = 1, 2,r1(w) : inf{t : X(w) + \u00C2\u00A3t(W) \u00E2\u0082\u00AC ,p = i,O,E:=i\Oo, B0:=0.Theorem 4.21 For each i assume (LIP). Then the process ict= (, \u00C2\u00A3) obtained by(4.74)\u00E2\u0080\u0094(\u00E2\u0080\u00994. 76) is a solution of the fixed point problem (4.18); i.e.I k \u00E2\u0080\u0094 = max [ (x + \u00E2\u0080\u0094 (x + K])](4.77) 1 - = [2 (x + - (x +for every t E [r\u00E2\u0080\u0099,T1) and any n E IN.Proof: The proof of (4.18) shows that (4.77) holds at least when T = T,j = 1,2; i.e.,when Kt is defined by (4.76). On the other hand, when = we have (cf. (4.74))(4.78) = krn = Vt E [T\u00E2\u0080\u0099,T\u00E2\u0080\u0099\u00E2\u0080\u0099);also, by the definition of r (cf. (4.75)8 with n instead of n + 1 and p = 3) followsx + e A\u00E2\u0080\u0099,which together with the definition of \u00E2\u0080\u0098r\u00E2\u0080\u00991 (cf. (4.74)4) implies(x + K, x + A\u00E2\u0080\u0099Chapter 4. The Optimal Control 129for every s e [T1, T\u00E2\u0080\u0099). Hence the RHS of (4.77) is zero; i.e., (4.77) holds.It remains only to show that (4.77) holds when T\u00E2\u0080\u00992 = T\u00E2\u0080\u0099; that is, when k is definedby (4.75). In this case let i e IN, i = i(i), be such that (cf. (4.75)7)n+l =and let t T\u00E2\u0080\u00991). First of all, we assume t < r\u00E2\u0080\u0099\u00E2\u0080\u00991; then (cf. (4.67) with r1\u00E2\u0080\u0099 instead ofr\u00E2\u0080\u0099) k \u00E2\u0080\u0094 = \u00E2\u0080\u0094 n = Also,[ (x + \u00E2\u0080\u0094 (x + \u00E2\u0080\u0098)J += max [ (x + + (k \u00E2\u0080\u0094 )) \u00E2\u0080\u0094 (x + i\u00E2\u0080\u0099)] +r (x + - (x +for every s e [flP, t]; hence, because of the induction hypothesis,\u00E2\u0080\u0094 > \u00E2\u0080\u0098 (x + \u00E2\u0080\u0094 (x +for every s E tj. Hence, in particular,>Therefore,X,P int(A\u00E2\u0080\u0099 U 1?2)and hence (cf. (4.75)9)T\u00E2\u0080\u0099 0, and at t = 0 when r1 = 0. Now we assume that (4.81)holds for all t e [0,r) and then we show that (4.81) holds also for all t\u00E2\u0082\u00AC[TZ,Tl).Let t e [,T1), then (4.77) implies(4.82)\u00E2\u0080\u0094= [ (x + \u00E2\u0080\u0094 (x +If the max above is zero (this includes the cases = r and r\u00E2\u0080\u0099 = Ti\u00E2\u0080\u0099), then by inductionwe have== [ (x + \u00E2\u0080\u0094 x];butVs\u00E2\u0082\u00AC[r\u00E2\u0080\u0099,t] :(since the max above is zero), hence= ma [ (x + \u00E2\u0080\u0094 X\u00E2\u0080\u0099];i.e., (4.81) holds for t. On the other hand, when = or then we can drop \u00E2\u0080\u009C+\u00E2\u0080\u009Cin (4.82). Therefore, from (4.82) we getJ = [7 (x + \u00E2\u0080\u0094 x8\u00E2\u0080\u0099] = [(x + \u00E2\u0080\u0094I It\u00E2\u0080\u0099In_.Hence the induction hypothesis implies=[k (x + \u00E2\u0080\u0094 x\u00E2\u0080\u0099],i.e.,= max [ (x + k) \u00E2\u0080\u0094 x]and we obtain (4.81). Similar arguments provide us with (4.81)2 and hence the proof ofthe Corollary is complete. \u00E2\u0080\u00A2Chapter 4. The Optimal Control 132Corollary 4.23 Assume (LIP) for both i = 1, 2. Then the process= (, ) obtainedby (4. 74)\u00E2\u0080\u0094(4. 76) is uniquely defined, adapted and cadlag, with non-negative and non-decreasing components. In particular, k is progressively measurable.Proof: This follows from (4.74)\u00E2\u0080\u0094(4.76) and Proposition 4.19. The last assertion in thestatement above is true for any process which is adapted and right-continuous. \u00E2\u0080\u00A2Remark 4.24 We believe that a unique solution of (4.55) exists even when (NIN) holds,or in the mixed case. However, since we have been unable to prove so, we assume (LIP)in order to have k adapted (cf. Proposition 4.19).It remains only to show that k as given by (4.74)\u00E2\u0080\u0094(4.76) is optimal for (x). We dothis by means of a generalized Ito formula for semimartingales and Sobolev functions.4.4 OptimalityLet the control k be defined by (4.74)\u00E2\u0080\u0094(4.76), then the corresponding controlled process:= X + k never leaves cl(A1) after time 0, i.e.(4.83) : t E (0,+oo)} C cl(A)as this follows from the construction of k (in fact, it suffices to argue as in the proof ofProposition 4.20 with the representation of k given by (4.81)). Notice that if b1 and \u00E2\u0080\u0098b2are continuous functions then the optimal process X is continuous too, except perhapsat time 0; in fact, the jumps of k (and hence of X) may only occur in correspondenceto the jumps ofRemark 4.25 It is clear that, because of (4.83), the optimality of k could be obtainedby means of the Ito formula for semimartingales and the dynamic programming equationChapter 4. The Optimal Control 133Alt = f (cf. Section 4.2). However, the classical formulation of the Ito formula requiresthe function ll (to which we would like to apply this formula) to be of class C2 at leastin a neighbourhood of cl(A\u00E2\u0080\u0099). But in the general setting of Chapter 2 we can only claimll e W\u00C2\u00B0(R2), and even under the more generous conditions of [WCM] (i.e., those ofSections 3.5 and 3.6) we are still unable to extend ll to aC2-function near the cornerpoint 0o The problem is that the presence of this corner prevents us from applyingthe classical PDE results about extension operators. This does not mean that \u00C3\u00B1 cannotbe smooth near 00, but only that a different approach is required (in fact, due to theparticular class V of admissible controls, it is very likely that the corner remains evenunder stronger conditions on the data of the control problem).A more general Ito formula was proved by Krylov (cf. [Kry], p 122); his result appliesto a non-degenerate diffusion in a bounded region and to functions having only generalizedsecond derivatives. We adjust Krylov\u00E2\u0080\u0099s theorem to account for the fact that our processX is a semimartingale.Theorem 4.26 Assume (LFP), (LIP), a non-degenerate, and &j continuous (i = 1,2).Then the process k = (k, ) defined by (4. 7)\u00E2\u0080\u0094(4. 76) is optimal for the original controlproblem (2.3); i.e.,(4.84) ll(x) = J() for allx 112Proof: Let x E 112 and let Xj = X + k be the controlled process starting at timet=Ofromx. Letm>ObesuchthatxO:IXtImorkm,iE{1,2}}and we apply Krylov\u00E2\u0080\u0099s version of the ItO formula to ll(f()e_Pt from 0 to r.Weset Bm = B(O,m(1+v\u00E2\u0080\u0099)) andwerecallthat (cf. Proposition2.2) ft E W2\u00E2\u0080\u0099\u00C2\u00B0\u00C2\u00B0(Bm+i),here Bm+i = B(0,m(1 + /) + 1); All = f a.e. in A\u00E2\u0080\u0099 (cf. (3.67)); also OA\u00E2\u0080\u0099I = 0 (cf.Chapter 4. The Optimal Control 134Proposition 3.37 and recall that .9A\u00E2\u0080\u0099 C \u00C3\u0094A\u00E2\u0080\u0099). Then we redefine All on the free boundary0A1; i.e. we set(4.85) All := f on .9A\u00E2\u0080\u0099.We approximate ll by smooth functions. Let us set Bm Cl(Bm). Then there existfl E C2(Bm) such that for q < 00IIu\u00E2\u0080\u0094 UIIC(m) ) 0,(4.86)\u00E2\u0080\u0094 VllhIC(m) 0IID2u \u00E2\u0080\u0094 D2ILIILQ(B ) 0;we now apply the Ito formula for semimartingales (cf. [My], p.278) to e_Ptu(f() from 0to r, hence we havee_PTu(Xr)\u00E2\u0080\u0094(4.87) = jm e (\u00E2\u0080\u0094All\u00E2\u0080\u0099)(t)ds + fm e 8Vuh1((s) dl5+ f eVu(X3)udW.Now we would like to pass to the limit in (4.87) and conclude an analogous result for ii.This is possible because of some very crucial estimates obtained by Krylov. Our setting ismuch simpler than his, since for us g and c are constant. Hence we trivially get estimateslike II K(det a)1!2 andE{Jrne8IgI ds} K(det a)\u00E2\u0080\u009D2j eds N(compare with {Kry], Lemma 2.5, p.54). However, in order to claim(4.88) E{Jrn e_P8Ih(s)Ids} Nhqfor any Borel function h(x) and with N = N(g, a, m, 2), we follow Krylov\u00E2\u0080\u0099s proof ofhis Lemma 2.8, p.56. All of Krylov\u00E2\u0080\u0099s arguments make use of \u00E2\u0080\u009Cmollification\u00E2\u0080\u009D and theChapter 4. The Optimal Control 135application of the Ito formula to smooth functions. In these formulae we make theopportune changes to account for the presence of k in the process X; i.e. , we add theextra termtArj eVh)(X3)dk8.But from the definition of T follows ktArA I 0,(4.90) M1 E (0,1) such that 1 \u00E2\u0080\u0094( Vf, (IvIi,\u00E2\u0080\u00941) 0,I Vf \u00E2\u0080\u00A2(1,M2)>0,(4.91) M E (0,1) such that 2 \u00E2\u0080\u0094( Vf.(\u00E2\u0080\u00941,M)0.Then we haveProposition 4.28 Assume (3.114), f e C3(1R2), (3.116), (4.90) and (4.91). Then jis a contraction (i = 1, 2); hence, in particular, (LIP) holds.Proof: This follows from [WCM], Theorem 4.12, and from the geometry of the regionof action. In fact, notice that (4.90) and (4.91) correspond to (3.170) (except for thefact that in (4.90) and in (4.91) we require M < 1 too). It is shown in [WCM] that,under (3.114), f e C3(R2), (3.116) and (3.170), one hasI Vv .(M1,1)>0,(4.92)\u00E2\u0080\u0094( Vv2 . (1,1v12) 0,Chapter 4. The Optimal Control 138where v is the solution of the penalized problem (3.123). (We point out that (4.92) isobtained by applying to A(Vv1 (Mi, 1)) and to A(Vv2 (1, M2)) a version of the maximum principle for elliptic operators andC2-functions bounded above by a polynomial;in fact, v6 e C4\u00E2\u0080\u0099(R2)if f E C3(R2).)Clearly (4.92)2, for example, means that v2 is non-decreasing along lines of slope M2;then, by using the fact that v2 \u00E2\u0080\u0094* uniformly on compacta as a \u00E2\u0080\u0094 0, we concludethat(4.93) T\u00E2\u0080\u00991(P) C Ro U R2 for every P e 02,where F1(P) is the cone with vertex F, defined by= {Q = (x?, x) : x? xi\u00E2\u0080\u0099, x x, (P\u00E2\u0080\u0094 Q) e2 P \u00E2\u0080\u0094 QI cos(arctan(1/M2))}if F = (xf, x) and \u00C3\u00AB2 is the unit vector in thex2-direction. (For the detailed proof ofthis see [WCM], Theorem 4.12).Now we apply these same arguments to our (4.90)2 and we obtain(4.94) Fr(P) C R U R2 for every P e 02,with= {Q = (xv, x) x? xf, x M2(x? \u00E2\u0080\u0094 xfl, therefore we have2(X1) > Xwhich contradicts P E 02 (i.e., P e graph(i,\u00E2\u0080\u00992)). Thus, the graph of 2 is constrainedto lie within the two cones with vertex F, with axis parallel to the x1-axis and angle2/ = 2 arctan(M). Moreover, since the same is true at every point of 02 (with the sameangle 2/3), we conclude that \u00C2\u00A3ip(2) M2, and hence, in particular, (LIP) holds. \u00E2\u0080\u00A2Remark 4.29 Notice that in Theorem 3.70 we had already shown that, if (3.114), f eC3 and (3.116) hold, then \u00E2\u0080\u0098i/\u00E2\u0080\u0099j is continuous everywhere and locally Lipschitz away fromthe corner point 0 (and hence i/ = Hence in Proposition 4.28, (4.90) and (4.91)allow us to improve Theorem 3.70 and obtain \u00C2\u00A3ip(i) < 1.Remark 4.30 We point out that in [WCM] the two extra conditions (4.90) and (4.91),with M > 0, (i.e., (3.170)) were introduced only to show that the points of the freeChapter 4. The Optimal Control 140boundary are points of positive Lebesgue density for the region of inaction, i.e. (4.93).However, this follows already from (3.114), f e C3(112)and (3.116) (cf. Section 3.6).Corollary 4.31 Assume (3.114), f e C3(1R2), (3.116), and(4.98) f12 0 in It2Then b1 is non-increasing (i = 1,2); hence, in particular, (NIN) holds. If in addition(4.90)2 (resp. (4.91)2) holds, then \u00E2\u0080\u0098b1 (resp. 1\u00E2\u0080\u00992) is also a contraction, and hence (LIP)holds for \u00E2\u0080\u0098/\u00E2\u0080\u0098i (resp. \u00E2\u0080\u00982).Proof: It is clear that (4.98) is a limiting case of (4.90), and hence the arguments ofthe proof of Proposition 4.28 apply and allow us to conclude that(4.99) P \u00E2\u0080\u0094 A* C (A for every P Ewhere A* = {(x1,x2) E 112 : x1 0, x2 0}. This implies that \u00E2\u0080\u0098/j cannot have anypoint of strict increase; i.e. /\u00E2\u0080\u0098j is non-increasing. So, in particular, (NIN) holds. The laststatement of the Corollary follows from Proposition 4.28. \u00E2\u0080\u00A2Notice that (by convexity) the condition (4.98) implies that (4.90) and (4.91) holdfor every M1,2 E 11+.Examples: Let be fixed constant, then(x1\u00E2\u0080\u0094i)2 (x2 \u00E2\u0080\u0094 x2)(a) f(xi,x2)= a2 + b2satisfies f12 0, hence the functions 4j are constant (in fact, b(x) \u00E2\u0080\u0098\u00E2\u0080\u0098(x), i j, if(x, 4) is the corner point). Therefore, the region of inaction A1 is just a translation ofthe first quadrant. Obviously both (LIP) and (NIN) hold.\u00E2\u0080\u0094 \u00E2\u0080\u0098) + ,i3(x2 \u00E2\u0080\u0094 x2)) (\u00E2\u0080\u0094/3(xi\u00E2\u0080\u0094 \u00E2\u0080\u0098) + c(x2 \u00E2\u0080\u0094(b) f(xi,x2)= a2 + b2Chapter 4. The Optimal Control 141satisfies (4.98) and (4.90)2 ifb2\u00E2\u0080\u0094 2()2 + (16a)2If this is the case, then the functions \u00E2\u0080\u0098i/ are non-increasing contractions. Both (LIP)and (NIN) hold. However, if K > 1 then the functions are only non-increasing and(NIN) holds. On the other hand, if K < 0 then and 2 are non-decreasing. Finally,if K e (\u00E2\u0080\u00941,0) then b1 and 2 are non-decreasing contractions, and hence (LIP) holds.(c) f(xi,x2)= (a(xi \u00E2\u0080\u0094 ,)2 +i3(x2 \u00E2\u0080\u0094 X2)T, r E IN,r 2,satisfies (4.90) and (4.91) if__12cj2V/32 2r\u00E2\u0080\u00941In particular, if o = ,6 = 1 then (4.101) is verified for r = 2,3, and (LIP) holds.We conclude this Section by showing that either of the two conditions (LIP) and(NIN) (together with the assumptions of Section 3.6) impliesC2-regularity of the valuefunctions \u00C3\u00B1 in the closure of A\u00E2\u0080\u0099, and so improves and completes Corollary 3.72.Proposition 4.32 Assume (3.114), f C3(1R2) and (3.116). For each i assume either(LIP) or (NIN). Then the free boundary is Lipschitz, i.e.(4.102) DA\u00E2\u0080\u0099 is of class C\u00C2\u00B0\u00E2\u0080\u009D.Proof: Because of Theorem 3.71,(ii), it suffices to analyze the regularity of 0A1 nearthe corner point Oo = (0, 0). There are three cases(a) both , and \u00E2\u0080\u00984&2 satisfy (NIN);(b) both , and 2 satisfy (LIP);(c) /\u00E2\u0080\u0098 satisfies (NIN) and satisfies (LIP), i j.In any of these cases it is possible to find a small neighbourhood Be(O\u00C3\u00BC) of 0o suchChapter 4. The Optimal Control 142that B(D0) fl DA\u00E2\u0080\u0099 is the graph of a Lipschitz map, since /ij is a function of x, i j,graph(,) fl graph(2)= O and \u00C2\u00A3ip(\u00E2\u0080\u0099/\u00E2\u0080\u0099jI[O,)) < 1. In fact, let lo be the straight linethrough and orthogonal to v = (+1, +i)/v; letp: B(\u00C3\u00B4o) fl Oil\u00E2\u0080\u0099 \u00E2\u0080\u0094*be the orthogonal projection of Be(O\u00C3\u00BC) fl OA\u00E2\u0080\u0099 on 10. Then P(Oo) = 0 and p is 1-to-i if Eis sufficiently small. Let ,u = (+1, \u00E2\u0080\u0094i)/i/ and define\u00E2\u0080\u0098ir :p(B(Oo)flOA\u00E2\u0080\u0099)\u00E2\u0080\u0094*Rby setting7r(p(P)) = p(P)Clearly ir o p is 1-to-i; then let h denote the inverse function of ir op. It follows thath(z2) \u00E2\u0080\u0094 h(z,) fi in case (a),\u00E2\u0080\u0094 Z1\u00E2\u0080\u0094 ( C(/3) in cases (b) and (c),where /3 := \u00C2\u00A3p(?/\u00E2\u0080\u0099lI[O,a)) VCP(\u00E2\u0080\u0099b2I[O,a)) and C(/3) is a constant which depends on \u00E2\u0080\u00983. HenceDA\u00E2\u0080\u0099 is Lipschitz. \u00E2\u0080\u00A2Corollary 4.33 Assume (3.114), f e C3(1R2) and (3.116). For each i assume either(LIP) or (NIN). Then the value function i2 is of class C2\u00E2\u0080\u009D in A\u00E2\u0080\u0099 up to the boundary,and hence(4.103) C2(cl(A\u00E2\u0080\u0099)).Proof: This follows from Proposition 4.32 by the application of a classical Sobolevembedding theorem for Lipschitz domains (cf [GT], Theorem 7.26, p. 171). \u00E2\u0080\u00A2Now, a posteriori, we can see how similar our result is to the one obtained by Davisand Norman [DN], in the sense that our optimal cost fz and their optimal profit show theChapter 4. The Optimal Control 143same behaviour outside of the region of inaction A1; moreover, although our A\u00E2\u0080\u0099 is notgenerally wedge-shaped, the direction of the reflection on the two branches 01, 0 of OA\u00E2\u0080\u0099turns out to be determined in a way analogous to the one in [DN]. This similarity of thetwo models is quite surprising if one observes that Davis and Norman allow the diffusion tohave state-dependent coefficients, and take as controls the processes of bounded variationof the formI\u00E2\u0080\u00941 \u00E2\u0080\u00941)Ut(here ). e [0, ), e [0, 1], and L, U are non-decreasing processes with L0 = 0 = U0).However, they restrict themselves to a very special form of the cost rate f in order todeduce their deus ex machina: the \u00E2\u0080\u009Chomothetic property\u00E2\u0080\u009D. On the other hand, our diffusion has constant coefficients, and our controls are monotone non-decreasing (componentwise), but the cost rate f is quite general, although convex. Nevertheless, there is onething common to both models, that is the cheap control setting. Then, perhaps, whatwe have obtained is a characterization of the class of cheap control problems, monotoneor not.Bibliography{Bd} F.M. 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"Geometric approach to monotone stochastic control"@en . "Text"@en . "http://hdl.handle.net/2429/3236"@en .