{"http:\/\/dx.doi.org\/10.14288\/1.0080232":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Lemieux, Marc","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2010-04-20T23:49:21Z","type":"literal","lang":"en"},{"value":"1983","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Let X be a semi-martingale. Techniques of [1] and El Karoui (in [3] and [4]) are used to study the convergence of 2\u220a times the number of upcrossings of [x, x+\u220a] by X to its local time at x. If X is continuous and if there exists a bicontinuous version of its local time process, then off a single null set, the convergence is shown to be uniform in x (and time). If X is such that the sum of the absolute value of its jumps over any finite time interval is almost surely finite, then, off a single null set, the convergence holds at all but countably many x. A notion of generalized arc length, is introduced, in the spirit of the quadratic arc length of [1], and the last result above is used to show that is the almost sure arc length of X, a uniform limit recoverable from the geometry of the trajectories.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/23964?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"ON THE QUADRATIC VARIATION OF SEMI-MARTINGALES by MARC LEMIEUX B . S c , M c G i l l U n i v e r s i t y , 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1983 \u2022 Marc Lemieux, 1983 ) I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 D a t e 7 DE-6 (3\/81) - i i -ABSTRACT Let X be a semi^martingale, Techniques of [1] and E l Karoui ( in [3] and [4]) are used to study the convergence of 2\u00a3 times the number of upcrossings of [x, x+e] by X to i t s l o c a l time at x . I f X i s continuous and i f there ex i s t s a bicontinuous ver s ion of i t s l o c a l time process , then of f a s i n g l e n u l l se t , the convergence i s shown to be uniform i n \u201e (and t ime) . I f X i s such that the sum of the absolute value of i t s jumps over any f i n i t e time i n t e r v a l i s almost sure ly f i n i t e , then, o f f a s i n g l e n u l l se t , the convergence holds at a l l but countably many x . A not ion of genera l ized arc length, i s introduced , i n the s p i r i t of the quadrat ic arc length of [1 ] , and the l a s t r e s u l t above i s c c used to show that i s the almost sure arc length of X , a uniform l i m i t recoverable from the geometry of the t r a j e c t o r i e s . - i i i -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i ACKNOWLEDGEMENT iv CHAPTER I Pre l iminary cons iderat ions 1. Presentat ion of r e s u l t s 1 - 3 2. Basic notions 3 - 6 CHAPTER II The upcrossings of a semi-martingale 1. A stopping time representat ion 6 -13 2. Uniform convergence to l o c a l time 13-20 CHAPTER I I I General ized arc length for semi-martingales 1. D e f i n i t i o n s , 21-2. A s p e c i a l case 21-23 3. The jump case , , , 23-37 BIBLIOGRAPHY. 38 - i v -ACKNOWLEDGEMENT Ed Perkins, the supervisor of t h i s work, deserves c r e d i t and thanks for h i s c l e a r understanding of i n f i n i t e s i m a l matters the e f f o r t he invested i n sharing i t with me. - 1 -I - PRELIMINARY CONSIDERATIONS 1. Presentat ion of r e s u l t s For a process X and a p a r t i t i o n Q of [ 0 , t ] , consider 2 the sum S(X,Q) = Z (X - X ) It i s well-known (for example, t \u20acQ i+1 t i i see 5, p . 355 ) that i f X i s a semi-martingale, then S(X, Q^) -> [X,X] i n p r o b a b i l i t y and L''\" whenever the Q 's are success ive t n d i v i s i o n s of [ 0 , t ] whose mesh goes to zero as n goes to i n f i n i t y . There i s no almost sure r e s u l t of t h i s k ind for a r b i t r a r y semi-mart ingales , but when B i s Brownian motion, i t i s known that S(B, Q^) -> t a . s . provided the Q n ' s a r e nested p a r t i t i o n s of [ 0 , t ] [2, p. 395]. The convergence of S(X,Q) uniformly over a l l p a r t i t i o n s of small mesh f a i l s : Taylor [7] has shown that sup S(B,Q) = \u00b0\u00b0 a . s . In f a c t , he showed that sup I f (B - B ) i s i n f m i t e i f r Q t . e Q fcl+l t i A * , 2 i s a funct ion such that log log (s) f (s) \/ s ->\u2022 \u00b0\u00b0 as s 0 , where log * ( s ) = max( l , | l o g ( s ) | ) , and i n [6 ] , Munroe has used t h i s r e s u l t to construct a square-integrable mart ingale M and nexted p a r t i t i o n s Q n of [ 0 , t ] such that sup S(M,Qn) = \u00b0o . This shows how the quadrat ic v a r i a t i o n approach i s inadequate i n i d e n t i f y i n g [X,X] as a uniform (or almost sure) l i m i t for a r b i t r a r y semi-martingales . There i s , however, another way to look at t h i s problem: i n C l ] , the t r a v e r s a l time for sect ions of path of Brownian motion was computed only us ing information a v a i l a b l e from the geometry of the t r a j e c t o r i e s . The t r a v e r s a l - 2 -time (or quadratic v a r i a t i o n , f o r Brownian motion) was recovered as an almost sure uniform l i m i t from the range of the process, independantly of any time parametrization; more p r e c i s e l y , i t was shown that f o r almost a l l sample paths, t i s the uniform l i m i t over a l l p a r t i t i o n s Q = (x^ : i^Z) of mesh less than e as e goes to zero of: (*) I(x_+i ~x_) x o f crossings of [x\u00b1, x _ + _ 3 b v B o v e r l\".0,t_) For Brownian motion, t h i s provides a d e f i n i t i o n of a generalized arc length, equal for almost a l l sample paths to the t r a v e r s a l time, independantly of p a r t i c u l a r parametrizations. This thesis i s an extension of t h i s state-space approach to the case of semi-martingales. The main r e s u l t i s theorem ( I I I . 3 . 3 ) ; i t shows that c c c the process , where X denotes the continuous martingale part of X , i s the uniform l i m i t , over a l l p a r t i t i o n s of the state-space, of sums of type (*) f o r a broad c l a s s of semi-martingales. Let X be a semi-martingale whose sum of absolute jumps before time t i s almost surely f i n i t e f o r each t . In [ 4 ] , E l Karoui has shown that the l o c a l time L^ _ of X i s , f o r each x , the almost sure l i m i t , as e goes to zero, of e times the number of upcrossings of the i n t e r v a l Dx,x+e] by X up to time t . Theorem ( I I I . 3 . 3 ) complements t h i s r e s u l t by showing that - 3 -the n u l l se t s , which depend on x , can be combined in to a s i n g l e one of f which the convergence holds at a l l but countably many x . The proof of ( I I I .3 .3) combines a g r i d argument of [1] and a stopping time representat ion for the upcrossings due to E l Karoui [43; both are presented i n Chapter I I , where theorem ( I I .2 .4 ) genera l izes a r e s u l t of Perkins [1 , thm. 2 ] : we show that i f a continuous semi-martingale has a bicontinuous v e r s i o n of i t s v e r s i o n of i t s l o c a l time process , then o f f a s i n g l e n u l l se t , the convergence of e times the number of upcrossings of [x, x+e] i s uniform i n x and t . Th i s provides a new g l o b a l c h a r a c t e r i z a t i o n of the l o c a l time associated with these semi-martingales . 2. Basic notions I t i s now time to introduce some conventions, and r e c a l l b r i e f l y the d e f i n i t i o n s and elementary r e s u l t s per t inent to t h i s study. We give ourselves a f i l t e r e d p r o b a b i l i t y space ( f i , F , F t , P ) s a t i s f y i n g the usua l hypotheses on which a l l processes considered i n the next chapters are de f ined . A semi-martingale X i s any adapted process that can be w r i t t e n as the sum of a l o c a l mart ingale M and an adapted process V of bounded v a r i a t i o n on the compact subsets of R + . We w i l l denote the d i s c o n t i n u i t y jumps X g - X g _ by & X s > | d v s | w i l l denote the t o t a l v a r i a t i o n measure induced by V on the Bore l sets , and X w i l l denote the continuous mart ingale part of X . A semi-martingale i s c a l l e d s p e c i a l i f i t admits a representat ion M + V as above where, i n a d d i t i o n , V i s p r e d i c t a b l e ; because of p r e d i c t a b i l i t y , t h i s representation i s unique, and M + V i s c a l l e d the canonical decomposition of X . A semi-martingale i s s p e c i a l i f and only i f the t o t a l v a r i a t i o n of V i s l o c a l l y integrable, and a continuous semi-martingale i s s p e c i a l [ 5]. If X i s a s p e c i a l semi-martingale, we denote i l k k |dV_I) } and we l e t H be the space |X||, = E 1 \/ k { [ M , M ] k \/ 2 + ( of a l l X such that ||x|| < \u00ab . In [5], P.A. Meyer has defined the l o c a l time process associated to semi-martingales through Tanaka's formula. Namely, l o c a l time i s the continuous, adapted process L which s a t i s f i e s ( 1 . 2 . 1 ) \\ L X = ( X t - x ) + - ( X 0 - x ) t - - 1(X _>x)dX - \\x) + _ (X -x ) + l(X _ R , f(X )d = s ' s ( 1 . 2 . 2 ) 0 3R f ( x ) L X d x a.s. This l o c a l time process has been extensively studied. In p a r t i c u l a r , i n [9], Yor has considered the sample path co n t i n u i t y - 5 -properties of the processes L, Z and L ; he has shown that i f a semi-martingale X i s such that the sum of the absolute values of i t s jumps before time t i s almost surely f i n i t e f o r each t , then: (1.2.3) (a) there e x i s t s a v e r s i o n of everywhere cadlag i n (x,t) , (.b) there e x i s t s a version of L everywhere cadlag i n x and continuous i n t , Cc) the jumps of these processes are given by t ~ ^ t = 2 [ l ( X g _ = x ) d V s + I ( A X s ) l ( X s _ = x)] sdVs 0 t = 2[ 0 l ( X s _ = * ) d X s - I (AX s)l(X _ = x ) ] s ) P} s s < C (b-a) P||x|| V H-F i n a l l y , we record another r e s u l t of Yor; i t i s an easy consequence of the density of occupation formula and the continuity properties of l o c a l time: (1.2.5) (Yor, [9] ) : Let X be a semi-martingale such that \u2022_ |AX | i s f i n i t e a.s. f o r each t . Then f o r each t , s0 % t C \u201e C T X l(x = LT\" , f o r a l l x , a.s. s s t II - UPCROSSINGS OF A SEMI-MARTINGALE 1. A stopping time representation Throughout the remaining chapters, X denotes a semi-martingale, J(X) denotes the sum of d i s c o n t i n u i t y jumps _ AX and K(X) denotes the increasing process _ |AX | . I f S t s s s : X < a) T b ( s ) = i n f ( t > S a ( s ) ; X_ > b) , f o r n > 1 n n t S a(s) = i n f ( t > T b ,(s) : X\u201e < a) , f o r n > 2 n n\u20141 t a b As usual, i n f ((f)) = 0 , and we agree to write and a b for S (0) and T (0) r e s p e c t i v e l y . When there i s no possible n n confusion, we w i l l omit the superscripts a and b . We define the number of upcrossings of the i n t e r v a l [a,b] by X between times s and t as: N +(s,t,a,b) = \u00a3 1(T (s) l n S i m i l a r l y , we define the corresponding number of downcrossings as: N\"(s,t,a,b) = I 1(S (s) b) , and we set N(s,t,a,b) , n>2 n S the t o t a l number of crossings, as N +(s,t,a,b) + N~(s,t,a,b) . We w i l l modify the notation a l i t t l e b i t and write N\"'\"(s,t,I) (respectively N~(s,t,I) and N(.s,t,I)) instead of N +(s,t,a,b) (respectively N~(s,t,a,b) and N(s,t,a,b)) when i t i s more - 8 -convenient to do so. We are interes t e d i n i s o l a t i n g the pieces of a sample path that constitute an upcrossing from those that constitute a down-crossing, so we introduce the sets: U +(I) =U + (Ca,b]) = u ((S ,T ]] n n n>:l = { ( c o,s) e 0, x ]R+ ; there e x i s t s an n f o r which S (CJ) < s < T ((D) } n - n and tT(I) = U-(Ca,b]) = uC.CS ,T )) . i n n n>l = { (io,s) e Q, x]R + t there e x i s t s an n f o r which S (d i ) < s < T (co)} n n and we define D +(I) = D + (Ca,b]) ..and D _(I) = D\"(Ca,b]) as the complements of IT^CO and U - ( I ) r e s p e c t i v e l y . C l e a r l y , U+(I) , and hence D+(I) , are predictable, while U (I) and D~(I) are optional sets. The next proposition shows how these sets and stopping times given an expression f o r the number of upcrossing of an i n t e r v a l . Although i t s proof can be found i n [43, i t i s reproduced here f o r the sake of completeness, (II.1 . 1 ) PROPOSITION ( E l Karoui): Let X be a semi-martingale, and l e t I = [a ,b3 be an i n t e r v a l . Then: (b-a)N+(0,t,a,b) t = -(X Q-a)+ - 1(D +(I))1(X _>a)dX s s + (X -a)l(D-(I\u00bb - I (X - a ) l ( D _ ( I ) ) l ( X _=al T n V T n n - I (a-X )1(X_ >a>X\u201e )1(S l n n n + (b-a) I 1(X _l n n n Proof\u2022 The idea i s to express (b-a)l(T nXg for a l l n . Moreover, i f n n n n X < a , then S = 0 (and X = X = X ) , while i f X > a , U J_ J_ 1 then > 0 . From t h i s , i t follows that for n > 2 , (1) (b-a)l(T a>X_ )1(S a>Xc )l(S^ 1 , using the d e f i n i t i o n of D'(I) , we get: t (2) ( b-a^CO^.a . b ) = (a-X 0) - l(D+(I))dX s + ( X t - a ) l ( D _ ( I ) ) - I (X - b ) l ( X _**l n n n n - I (a-X )1(X >a>X\u201e )1(S l S n S n \" S n n - (a-X 0)l(X 0 and so the stochastic i n t e g r a l i s equal to 1(D+(I))1(X _>a)dX . Moreover, s s 0 i f we denote the set D +(I) n.{X _= a} by J(a) , then by (1.2.3), t S the i n t e g r a l l ( J ( a ) ) d X i s equal to _ (X :-a)l (J(a)) + sl T n T n ~ n The r e s u l t of these comments i s the following expression f o r the i n t e g r a l of (2): t (3) l ( D + ( I ) ) d X s = l(D+(I))l(X s_>a)dX c J + I (X s-a)l(D-( I))l(X s_=al n n + I (X -a)l(X\u201e =a)lCS l b n V n 12 V, Now, using simple properties, the two sums of (2) become; (4) I (a-X )1(X >a>X )1(S l n n n = I (a-X )1(X_ ^ K S ^ t ) n ^ l n n + I (a-X )1(X >a>X )1(S l n n n and (5) I (X T - b ) l ( X T _****l n n n = I (X T -b)l(al n n n = I (X T - a ) l ( X T _l n n n + (a-b) I 1(X T _^a****l n n Noting that Ca-XQ) - (a-X 0)l(X Q a ) d X s 0 + [ ( X t - a ) l ( D - ( I ) ) - ( X t - a ) \" r ] + E t(a,b) where (II.1 . 3 ) E t(a,b) = (b-a) _ 1(X T _l n n + C ._ (X - a ) l ( X a>X ) - I (a-X_ )1(X >a>X )1(S l S n S n \" S n n - I (X T -b)l(al n n n 2. Uniform convergence to l o c a l time Our aim i s now to show that, f o r c e r t a i n continuous semi-martingales, 2 (b-a)N (0,t,a,b) converges to (as b -I- a) , uniformly i n a and t , o f f a si n g l e n u l l set. This w i l l generalize, f o r these semi-martingales, what i s already known of Brownian motion [ 1 ] . - 1 4 -An easy argument using the d e f i n i t i o n of D~(I) shows that | ( X - a ) l ( D \" ( I ) ) - ( X t - a ) + | < (b-a) . Indeed, i f t e D~(I) , then X ^ a and the left-hand side i s zero. It i s also zero i f t | D~(I) and X < a . In the only case that remains, we have a < Xfc < b and t h i s makes the conclusion c l e a r . The following lemma gives conditions under which the Lebesgue-S t i e l t j e s part of the stochastic i n t e g r a l of (II.1.2) converges uniformly over the state-space: (II.2.1) LEMMA: Let X be a semi-martingale. (a) If X i s such that K(X) t i s f i n i t e a.s. f o r each t , then f o r each t , l i m l(x-ex)dV | = 0 a.s. s s o Proof of (a): F i x t , and pick CJ out of a n u l l set so that t K(X) and \/| dV | are f i n i t e . Then, by dominated convergence, 0 s' t lim e-s-0 l(x-ex) < l(x 0 t n uniformly, where f (x) = J l(x g(x) by dominated convergence, and i f y '+ x , then t g(y) converges to J l ( X 0 . QED. x n Remark: In the p a r t i c u l a r case of a continuous semi-martingale, t part (a) of the lemma says that a.s., |\/l(x-e 0 , the set t s {max sup xeR, t****1(X _>x)dX^| > k 1 \/ 2 i.o.} i s n u l l . Note: We may (and w i l l ) assume that the stochastic i n t e g r a l s t Y (x,k~ 6) = l ( u + ( C x , x + k \" 6 ] ) ) l ( X _>x)dX\u00b0 u s s J 0 are defined f o r a l l xe u R, o f f a sing l e n u l l set. k>l k Proof: By hypothesis, there e x i s t s a sequence of stopping times 2 (a : n > 1) such that a f 0 0 and X e H for a l l n . n n tAcr \/^c xn j \u201en, , -o, Denote by X^ , (X C)\u00b0 and Y^(x,k b) the processes X_ A , t -ft n x\u00a3 and Jl(u +([x,x+k ]))1(X n_>x)d(X\u00b0) n r e p s e c t i v e l y . Let n 0 B > 0 . Then, f o r any n , i t i s straightforward that: (1) P{max sup lY^x.k\" 6)! > k \" 1 \/ 2 i.o'.} xeR^. t**** k \" 1 \/ 2 i.6 .} + P{a < B} xeR k t**** k~ 1 \/ 2} xeRfc t**** x ) d < ( X C ) n > ) Z} where i n the l a s t l i n e , we used the Burkholder-Davis i n e q u a l i t y . Now use the fa c t that l(u + (Cx,x+k~ 6]))l(X n_>x) k i.o.} = 0 for any n . Pick n xeR. t**** B a.s.; the conclusion now follows (1). QED. (II.2.3) Remark: If X i s continuous, the error term E t(a,b) of (II.1.2) i s zero, and using the argument made i n the beginning of t h i s section, one finds that: t | (b-a)N+(0,t,a,b) < |J l (u+([a,b]))(X s_>a)dX g| + C(X t-a)l(D ([a,b])) - ( X t - a ) + ] l ( u +([a,b]))l(X _>a)dV I + s s Ku + ( C a , b ] ) ) l(X s> a)dM g 0 + (b-a) It follows from t h i s and both previous lemmas that for any 6 > 0 and each t such that 1\/ i s continuous, the maximum of - 18 -|k ^N +(0,t,x,x+k ^) - I o v e r the g r i d R^ Is bounded, o f f a \u20141\/2 \u20146 n u l l set, by 6 + k + k for large enough k . This i s the key fact needed to get uniform convergence of (II.1.2): (II.2.4) THEOREM: Let X be a continuous semi-martingale, and suppose a bicontinuous version of i t s l o c a l time process e x i s t s . Then, with p r o b a b i l i t y one, for any B > 0 , lim sup |eN +(0,t,x,x+e) - y L X | = 0 e-K) xeH t**** 0 , because Lj i s uniformly bicontinuous over [0,B] x ]R\" (for fixed OJ , i t has compact support i n the space n va r i a b l e ) we can p a r t i t i o n [0,B] in t o u [ t . , t . .^1 , where i = l i i+1 0 = t\u201e < t\u201e < ... -, t , < t = B i n such a way that f o r a l l 1 2 n-1 n -1 'i+1 I X X I L - L I < 6 . Then i f t \u00a3 [t.. , t ) we use the i t n - j . i t i 1 1 + 1 t r i a n g l e i n e q u a l i t y and the monotonicity of N~*\"(0 -,x,x+e) to obtain: | eN +(0,t,x,x+\u00a3) - y L X | < | eN+(0,t,x,x+\u00a3\") - \u00a3N +(0,t i,x,x+\u00a3) | + |\u00a3N +(0, t l,x,x+\u00a3) - | L X I + | | L X - | L X | i i < |\u00a3N +(0,t 1 + 1,x 1x+\u00a3) - y L X I + 2|eN+(0,t.,x,x+e) - y L X | + 5 i+1 i From the l a s t i n e q u a l i t y , the s u f f i c i e n c y of (*) i s apparent. - 19 -Proof of (*): F i x t > 0 , and l e t 0 < 6 < 1 . By remark (II. 2.3), we can select w out of a n u l l set, and N = N(o)) such that the following hold: (1) ( k - l ) ~ 6 - Ck-1)~7 > k\" 6 , for a l l k > N (2) sup |X (u))| + 1 < N s N x e R k F i x e.<:(N+l)~6 and l e t m > N+1 be such that (m+l)~ 6 < e < m~6 . If |x| > N - l , then L X = N +(0,t,x,x+e) = 0 . On the other hand, i f |x| < N - 1 , then define x^ = max(y e R^ : y < x) f o r a l l k > N . It i s easy to v e r i f y that: (4) x < x < x + e < x , + ( m - l ) ~ m-I m-1 (the l a s t i n e q u a l i t y follows from the choice of m >e (1) and the fac t that x - k 7 < x, .') and (5) x < X m + 2 + (m + 2)\"7 < X m + 2 + (m + 2)\"7 + (m + 2)\"6\" ?\u00bb xmf2+(m+2)-7 + (m+2)\"6] Then, (4) and (5), along with the monotonicity of I N +(0,t,I) - 20 -and the choice of m imply that (6) (nrf-l)~V(0,t,I _) - y L X m-l z t < eN+^t.x.x+e) - y L X \" m ( 0' t' Im+2 ) \" 2 L t Next, we note that: sup | (m+l)\" 6N +(0,t,I .) - y L X | xelR m _ 1 2 t < (m+l)\" 6(m-l) 6 max I (m-l)\"V(0,t,Y,Y+(m-l)\" 6 -kl ' YeR , Z m-l X ,,,,..-6, -.6 i l T m-l 1 T x I + (m+1) (m-l) sup | _ L f c ~2t ' xelR + C (m+1)\" 6(m-l) 6-1] sup | y L X | xelR By the uniform c o n t i n u i t y and boundedness of 1\/ , the l a s t two terms on the right-hand side go to zero as m + \u00b0\u00b0 ; therefore, by (3) f o r large enough m , i t i s majorized by 36 ... An i d e n t i c a l argument shows that sup lm 6 N + ( 0 , t , 1 ^ , \u201e ) - ^ - L X I < 36 for large m . xeJt This, i n view of (6), completes the proof. QED. 1 X I eN (0,t,x,x+e) - y L | -* 0 as xelR C e -> 0 because the differ e n c e between the number of upcrossings and down-crossings of a same i n t e r v a l i s at most one. - 21 -III - GENERALIZED ARC LENGTH FOR SEMI-MARTINGALES 1. D e f i n i t i o n s We now r e c a l l from Cl] the notion of arc length f o r stochastic processes. Throughout t h i s chapter, we w i l l denote by (x^ : i e ZZ) the points of a p a r t i t i o n Q of the r e a l l i n e , and ||Q|| w i l l denote the mesh of Q . We w i l l c a l l a p a r t i t i o n Q regular i f i t s a t i s f i e s x? < x ? M for a l l i , l i m x? = \u00bb , and l i m = -\u00bb . 1 l + l ' . x , i i -voo i - > - o o We w i l l denote by TT the c o l l e c t i o n of a l l regular p a r t i t i o n s of R , and for any v > 0 , we define i r ( v ) to be the subset of i r c o n s i s t i n g of p a r t i t i o n s of mesh no larger than v . If Q e ' . i r and X i s a process, we define: K^(X,Q) = I ( x J + 1 - x J ) 2 N ( s , t , x ^ x 2 + 1 ) i (III.1.1) DEFINITION: For any a e R and e ' > 0 , l e t P ( a , e ) denote the l a t t i c e {a+ie : ie2Z). We say a process X has arc length L (where L: R + -> R) i f for a l l 0**~~ 0 , li m sup I K ^ X . Q ) - ( - )| = 0 v+0 Qeir(v) s t s t~~** on [0,B] and the monotonicity of KQ(X,Q) and shows that .': i t s u f f i c e s to prove: (*) l i m sup |K^(X,Q) - | = 0 a.s. for each t v-K) Q\u00a3TT(V) Proof of (*): F i x t > 0 . By d e f i n i t i o n , we have that: KQ(X,Q) = (xj - x J ) N ( 0 , t , x ^ x J + 1 ) l [ X Q XQ ) ( x ) d x . Therefore, using the occupation time density formula (1.2.2), i t i s easy to see that: (1) sup | K'(X,Q) - ! QeirCv) sup |(x^ + 1-x^)N +(0,t,x^,x^ )1 ( x ) - | L X | d x Qeir(v) Cx?,x? L l) l l + l -- 23 -+ j sup |(xJ+1-\u00abJ)N-(0,t,xJ+1)l ( x ) - \u00b1 L X | d x E Q \u00a3 7 R ( V ) [ x i ' X i + l ) Now, observe that f or fixed CD , both integrands are of compact support, and that s u p i ^ - x ^ O . t . x J , ^ ) ! \u00ab-Kl ^ i ^ i + i ) xeJR x Q , sup Kx^ -xXw.t.xJ.x^ )-!^ 1 I + Qeu (v) 1 X - Q i \u2022 I 1 T 1 1 T X I s ? p q Q h L t - 2 L t l x\u20acLx^,x i + 1) Qeir(v) Theorem (II.2.4) and the uniform c o n t i n u i t y of show that the right-hand side of t h i s l a s t i n e q u a l i t y goes to zero with v . Thus, by dominated convergence, the right-hand side of (1) goes to zero with v , and t h i s completes the proof. QED. 3. The jump case The aim of t h i s section i s to generalize the r e s u l t of proposition (III.2.1) to semi-martingales X such that K(X) f c i s almost surely f i n i t e f o r each t . This extension i s a natural one; the almost sure convergence of K(X,Q) to depends on the convergence of 2(b-a)N +(0 ,t ,a to L as a + x and b .4- x , and t h i s , modulo countably many x where |dV | may charge the sets {X _ = x} , hinges on s s the behaviour of the error term E t(a,b) ,. But in proving that - 24 -+ a 2(b-a)N (0,t,a,b) converges to L a.s. as b 4- a i n the jump case [4], E l Karoui showed that E t(a,b) goes to zero a.s. as b 4- a . The next lemma shows that, i n f a c t , E^(a,b) goes to zero as a i x and b 4- x whenever x i s not i n the range of the d i s c o n t i n u i t i e s of the process: (III.3.1) LEMMA: Let X be a semi-martingale such that K(X) f c i s a.s. f i n i t e for each t . Then for each t , with p r o b a b i l i t y one, (a) (El Karoui) lim|E (x,x+e)| = 0 for each x e TR. . \u00a3+0 '(b) l i m sup [E t(a,b) = 0 at a l l but countably many e+0 Ca,b]c[ x-e,x+\u00a3] x e: TR . Proof of (a): F i x t > 0 , and pick u out of a n u l l set so that K(X) t i s f i n i t e . Consider the following increasing processes: A (x) = I (X - x ) l ( X _x>X ) s L' u u - u u**~~ 0 : E t(x,x+e)| < |A t(x) - I (X T - x ) l ( X T _l n n n - 25 -+ | B t ( x ) - _ (x-X s )1(X S _>x>Xg ) l ( s n < t ) n>l n n n + |C (x) - I (X -x)l(D ([x,x+\u00ab_,))l(X _=xl n n We w i l l show that each of the terms on the right-hand side of t h i s i n e q u a l i t y goes to zero with e . ( i ) Since every d i s c o n t i n u i t y jump of X from below x to above x+e x + e occurs at some T , using the d e f i n i t i o n of A (x) we n ' 6 s v f i n d that J (X T -x)l(X_ l n n n = I (X - x ) l ( X -x+e)dA ( x) s s Therefore, |A t(x) - _ - x ) l ( X T _l n n n 1(X 0 . Moreover, i t i s easy to see that the following i n e q u a l i t i e s hold: I (X T -(x+e))l-(xl n n . n < J (X -(x+e))l(xl n\" n n < e I 1(X _ s oo s Again, using dominated convergence shows that the right-hand side of both these i n e q u a l i t i e s goes to zero as t -> 0 . X+\u00a3 ( i i ) Using the d e f i n i t i o n of the stopping times , we f i n d that 1(X >x+e)dB (x) = 7 (x-X )1(X _>x+e>x>X ) s s ^ s v s s sx+e>x>X_ )1(S t> - b n n>i n n n < I (x-X )1(X >x>Xc )1(S \u00a3t) n>l S n S n ~ S n n < I (x-X )1(X >x>X ) = B. ( x ) sx>Xo )1(S l Sn S - \" S n n n KX s_ 0 , and pick to out of a n u l l set so that K(X) t i s f i n i t e . Choose x f' {X ,X : sSt,AX #)} , f i x \u00a3 > 0 and l e t a,b be any two r e a l s such that [a,b]c[x - e,x+ e ] . t t By d e f i n i t i o n of A (a) , Jl(X ~~**l n n n ^ It also follows from these c a l c u l a t i o n s that: t 1(x-el n n n l(x - ex+e)d(A+C) (x) shows that: s (3) (b-a) I KX^ _l n n - 28 -t t ( x _( x- e )) 1(X sSx+\u20ac)d(A+C) (x)+ 0 l(x-ea>Xr ) 1 ( S l b n b-~ s-' n n ~n 1 (x-e s 1 . Let L (X) and L (Y) denote r e s p e c t i v e l y the l o c a l time processes (at x) of X and Y . Then f o r each t , L X(X) = ( l + x 2 ) L ^ r C t s ( x ) ( Y ) f o r a l l x , a.s. 2 Proof: Since Arctg(\u00ab) i s a C function, i t follows from Ito's lemma that Y i s a semi-martingale, and moreover, K ( Y ) t i s f i n i t e a.s. for each, t because the d e r i v a t i v e of Arctg(-) i s bounded by 1 . The function Arctg(\u00ab) i s i t s e l f bounded by ir\/2 , and hence, Y i s l o c a l l y i n H P f o r any p > 1 . Let A(x,e) = Arctg(x+e) - Arctg(x) . Ito's lemma y i e l d s that dg = (l4X 2_)~ 2d s ; therefore, using the f a c t that c c Arctg(\u00ab) i s one-to-one and the f a c t that d g does not charge the sets {s 0 , (1+x 2) 2Ce\/A(x,e))e 1 t l(x s s ^ (A(x,e)) 1 0 t 1 (Arctg (x) g J 0 t > (l+(x+ e) 2) 2(\u00a3\/A(x,e)) e 1 J 0 l(x s s - 30 -Therefore, noting that e\/A(x,e) (1+x ) as e -* 0 and using lemma (1.2.5), taking l i m i t s i n the l a s t equation y i e l d s the desired r e s u l t . QED. We are now ready to extend the r e s u l t of (III.2.1) to semi-martingales with jumps. In the following pages, the Lebesgue measure of an i n t e r v a l I i s denoted by | l | . For any 'V > 0 and x e R , we define S(x,v) to be the c o l l e c t i o n of a l l i n t e r v a l s I such that x e I and | l | 0 , at a l l but countably many x e ]R . (b) l i m sup |Kfc(X,Q) - ( ^ )| = 0 v+0 Qeir(v) s t s t**** 0 . c c In p a r t i c u l a r , f o r a.a. sample paths, i s the arc length of X . Note: An argument using the uniform continuity of L on [0,B] and the monotonicity of N +(0,\u00ab,I) shows that i n order to prove (a) , i t s u f f i c e s to show that f o r each t , (a') lim sup | |l|N+(0,t,I) - y L X | = 0 v*0 IeS(x,v) C - 31 -at a l l but countably many x e R , a.s. In addition, by lemma (III.3.2) and since the number of upcrossings of any i n t e r v a l Ca,b] by X i s equal to the number of upcrossings of [Arctg(a), Arctg(b)3 by Y=Arctg(X) , we can assume that X 2 i s l o c a l l y i n H . This assumption only serves i n proving (a) ; the proof of (b) uses the r e s u l t of (a) and does not require that X be 2 l o c a l l y i n H any further. R e c a l l from the proof of (III.2.1) . that i n order to prove (b), I t s u f f i c e s to assume that s = 0 and to show that f o r each t : (b') l i m sup |(K^(X,Q) - | = 0 a.s. QeTT(v) Proof: F i x t > 0 and l e t 0 < 6 < h Define the sets D (co) = {X (cb), X _(.co) : s-0 [a,b]cCx-e ,x+e] x \\ D and max | J l (u+([x,x+k~ 6])) 1(X _>x)dM |T such that the following hold: - 32 -(1) 1(x-ex)dM | < k 1 \/ 2 for a l l k > N X \u00a3 R k O (4) I\"J-^ t ~ \" i ^ t ^ < ^ whenever | y-x | < \u00a3 (5) sup |X (o))|+KN (6) s k 6 f o r k > N The f i r s t step i s the set-up of the g r i d approximation: pick \u2014fi v < (N+1) and I e S(x,v) . Then by (8) and the d e f i n i t i o n of S(x,y) , we have that I c [x-\u00a3\/8, x + \u00a3 \/ 8 ] . Denote the i n t e r v a l I by [a,b] . Now, pick m = m(I) > N + 1 such that (m+1) 6 < | l | < m~6 . I f |x| > N - l , then L X = N+(0,t,I) = 0 and the theorem holds t r i v i a l l y . On the other hand, i f |x| < N - l , def ine: a^ = a^CO = max(y \u00a3 : y < a) for a l l k > N , and set Xm-1 = C am-1' a m - l + ( m - 1 ) \" ] and Xm+2 \" [ am+2 + ( n H- 2 ) _ 7> a m f 2+(m+2)- 7+(m+2)- 6] - 33 -As i n the proof of (II.2.4), i t i s an easy exercise to v e r i f y that m+2 m-1 Therefore, using the monotonicity of N +(0,t,\u00bb) and the choice of m , one finds that: (10) sup | |l|N+(0,t,I) -IeS(x,v) t < max{ sup |(m+l) 6 N + ( 0 , t , I .) - ^ L X | IeS(x,v) m - 1 2 t' sup |m V(0,t,I\u00b0 ) - ^ L X|} l\u00a3S(x,v) m + 2 2 t To complete the proof of (a'), we w i l l show that the choice of e and N makes the right-hand side of (10) small. We w i l l also prepare the proof of ( V ) by bounding (10) independently of x . Using (6) and the f a c t (which follows from (7)) that |(m+l) 6(m-l)~ 6 - l | < 6\/B < 6 for a l l m > N + 1 , one obtains (11) sup | (M+l)\"V(0,t,I ) - |-L X| IeS(x,v) m - 1 1 fc IeS( sup | (m-1) V ( 0 , t , I ) - j-L + (1+6) sup \\\\*^X-\\\u00a3 \\ + ( V B ) | f L X IeS(x,v) Now, i t follows from (8) and the d e f i n i t i o n of a, that k ! a m - l \" x l < e \/ 2 f o r a 1 1 m - N + 1 ' Therefore, by (4), 1 am\u20141 1 x sup | \u2014 L ., ~ \" 9 L ^ I < 5 because the sup i s taken over m>N+l IeS(x,v) 1 t 1 t - 34 -Moreover, using (II.1.2) and (8), (12) sup | (m-l) V ( 0 , t , I n ) - ^ L * IeS(x.v) m _ 1 2 t t -6 < N + sup 1 ( a m - l < V \" V l + (m-l)6)U(V+J)! IeS(x,v)\u00a3 t + sup I l ( u + ( I 1 ) ) 1 ( X _ > am )dM x o \/ \\ J m-l s m-l IeS(x,v) ^ + sup |E (a a m + (m-l) 6 ) | IeS(x,v) Observe that f o r a l l m> N + 1, the: i n t e r v a l s I , are contained m-l i n [x-e, x+e] (t h i s follows from (8) and a c a l c u l a t i o n above). Therefore, using (1), (2), (3) and (7), we see that the right-hand l T x aea D y to . iNocxng mat | equation (11) now shows that: 1 x side of equation (12) i s bounded by 46 . Noting that \u2014 L < B , sup | (m+l)\" 6N +(0,t,I ) - ^ L X | < H5 l\u20acS(x,v) m 1 t An i d e n t i c a l argument show that: sup |m\"V(0,t,I\u00b0._) - ^ L X | < 116 IeS(x,v) m + 2 2 t and the conclusion of (a') now follows from (10) and the fac t that the sets and are countable. The proof of (a ) i s complete, To bound the right-hand side of (.12) independently of x , r e c a l l that i n the l a s t l i n e s of the proof of lemma (III.3.1)(b) i t i s shown that sup |E t(a,b)| < 8'K(X) . [a,b]c[ x-e,x+e] - 35 -Therefore, from equations (6) and (12), using the l a s t remark, we get that f o r a l l x , sup ] (m-l)~ 6N +(0,t,I -) - y L X | < cB IeS(x,v) m 1 t where c i s a constant independent of x . Again, an i d e n t i c a l argument shows a s i m i l a r bound for the upcrossings of , and m+z so, i n view of (10), for a l l x , sup | | l['N +(0 ,t ,T) - j L X | IeS(x,v) < c B. To prove (b'), f i r s t note that i t follows from part (a) that: (13) l i m \\H-0 3R sup | |l|N(0,t,I) - L X|dx = 0 IeS(x.v) Z because the integrand has compact support, i s bounded independently of x and converges to zero at a l l but countably many x e K. . Consider the interval-valued functions f~(x) = [x?,x?,.,)l - n (x), Q I l + l r Q Q s Cx.,x i + 1) and r e c a l l that from the d e f i n i t i o n of KQ(X,Q) and the density of occupation time formula we have: \u00bb |KQ(X,Q) - :J < sup I |f (x)|N(0,t,f (x)) - L X f d x i Q6TT(V) Now, i f Qeir(v) , then f q ( x ) \u20ac 5(x,v) for each x , and therefore, (13) implies that the right-hand side of the above in e q u a l i t y goes to zero with v . This f i n i s h e s the proof of the theorem. QED. The argument of the proof of (a) above can be used to obtain a stronger version of theorem (II.2.4); incorporating i n the proof of (11.24) the set-up of i n t e r v a l s I^,\u201e c I c I c [x-e,x+e] m+z m-l for each x and using the uniform convergence of the Lebesgue-S t i e l t j e s i n t e g r a l s of (II.2.1) shows that i f X i s a continuous - 36 -semi-martingale whose l o c a l time i s bicontinuous, then a.s., for any B > 0 , lim sup sup | |l|N +(0,t,I) - y L X ] = 0 v-K) xeIR IeS(x,v) t t**** , can be used to i n t e r p r e t some stochastic i n t e g r a l s as uniform l i m i t s of Lebesgue i n t e g r a l s . Let X be such that K(X)^ i s a.s. f i n i t e for each t , and for the sake of argument, l e t g be a continuous function. Then, the dominated convergence argument i n the proof t of (III.3.3)(b) y i e l d s that \/ g(X )d = \/g(x)L Xdx = 0 S S R lim sup \/ g ( x ) ( x ^ + 1 - x 1 ) N ( 0 , t , x ^ , x ^ + 1 ) l Q Q (x)dx . Therefore, v+0 Qen-(v) R. [x^ \u00bb x ^ + 1 ) i f f e C^(R) and F' = f , an a p p l i c a t i o n of Ito's lemma gives that \/ f (X )dX c = F ( X J - F ( X ) - i l i m sup 'Jf' (x) (xj. -x?) 0 S S Z C 1 v+0 Qp(v) K 1 + 1 1 N ( 0 , t , x ^ , x ^ + 1 ) l ,Q Q (x)dx and so, (III.3.3)(b) can be used to rx.,x 1 + 1) recover some stochastic i n t e g r a l s through the geometry of the paths. - 38 -BIBLIOGRAPHY [1] Chacon, R.V. et a l . Generalised arc length for Brownian motion and levy processes, Z.W. Gebiete, Vol. 57 (1981), pp. 197-211. ~-[2] Doob, Stochastic processes, Wiley (1953) [3] E l Karoai, N. Sur l e s mont\u00a3es des semi-martingales (CAS continu ), Asterisque, Vol. 52-53 (1977), pp. 63-72. [4] E l Karoui, N. Sur l e s mont\u00a3es des semi-martingales (CAS discontinu ), Asterisque, Vol. 52-53 (1977, pp. 73-88. [5] Meyer, P.-A. Un cours sur l e s integrales stochastiques, Lecture notes i n math, Vol. 5111, Springer-Verlag (1976). [6] Monroe, I. The quadratic v a r i a t i o n of martingales: A counter-example, Annals of P r o b a b i l i t y , Vol. 4 (1977), pp. 133-138. [7 3 Taylor, S.J. Exact asymptotic estimates of Brownian path v a r i a t i o n , Duke Math Journal, V o l. 39 (1972), pp. 219-241. [8] Walsh, J. A d i f f u s i o n with a discontinuous l o c a l time, Asterisque, Vol. 52-53 (1977), pp. 37-46. [9] Yor, M. Sur l a con t i n u i t y des temps locaux associ\u00a3s A certaines semi-martingales, Asterisque, Vol. 52-53 (1977), pp. 23-36. 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