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On the quadratic variation of semi-martingales Lemieux, Marc 1983

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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 • 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£ 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 „ (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 <X , X > 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 €Q 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) = °° 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 ->• °° 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) = °o . 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±, 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 <X , X > , 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|| < « . 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 - \<L? s s 2 t where \l* = _ ( x -x) 1(X >x) + _ (X -x ) + l(X _<x) s<t S s<t S S and we define LA = L A + Ji^ t t t Meyer has also shown that the l o c a l time process s a t i s f i e s a density of occupation formula: for any p o s i t i v e , Borel measurable function f : R + -> R , f(X )d <X C,X C> = 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)] s<t L X - L X = 2 t -t "J 1 (V = X>dVs 0 t = 2[ 0 l ( X s _ = * ) d X s - I (AX s)l(X _ = x ) ] s<t s We w i l l always assume that we are working with these versions whenever conditions permit. Note that the l a s t formula i n (c) implies that a bicontinuous v e r s i o n of L e x i s t s i f and only i f the measure | dV | does not charge the l e v e l sets '{s: X _ = x} , s s f o r any x e R . I t also shows that the stochastic i n t e g r a l t / l ( X _ = x)dX i s the sum of £ AX 1(X _ = x) and a continuous 0 S S s<t S S process of bounded v a r i a t i o n on compacts, of support contained i n {s : X _ = x} . s The proof of (1.2.3) r e l i e s on an estimate f o r a p a r t i c u l a r - 6 -type of stochastic i n t e g r a l s . We record i t below, as i t w i l l be quite h e l p f u l i n handling i n t e g r a l s which a r i s e from the upcrossing representation: (1.2.4) (Yor, [9]): If X e H P , then there e x i s t s a constant c < °° such that: P f 2P | K a < X <b)dX^_|| 2 = E{sup( 1 s s H p t J 0 0 l ( a < X _ < b ) d < X ° , X c > ) 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 •_ |AX | i s f i n i t e a.s. f o r each t . Then f o r each t , s<t S l i m (h e>0 % t C „ C T X l(x<X_ <x+e)d< X.X1- > = 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<t s<t i t i s s p e c i f i e d that K(X) t i s almost surely f i n i t e f o r each t , then we write M + V f o r the canonical decomposition of the continuous semi-martingale X - J : note that i n t h i s case, the representation X = M + V + J i s unique, but i f , i n addition, X i s i n f o r some p , M + V + J i s not, i n general, the canonical decomposition of X . Let I = [a,b] be an i n t e r v a l . We describe the successive entry and e x i t times of I by X a f t e r time s with the following stopping times: S^(s) = i n f ( t > 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„ < a) , f o r n > 2 n n—1 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) = £ 1(T (s) <t) . n>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) <t) + 1(X >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» - I (X - a ) l ( D _ ( I ) ) l ( X _=a<X ) s<t S s s - I (X T -b)l(a<X T _<b<XT ) l ( T n < t ) n 1 n n n - I (X - a ) l ( X <a<b<XT )1(T <t) n>l T n V T n n - I (a-X )1(X_ >a>X„ )1(S <t) -, b b - b n n>l n n n + (b-a) I 1(X _<a<b<X )1(T <t) n>l n n n Proof• The idea i s to express (b-a)l(T n<t) using X T A and n X 0 . By d e f i n i t i o n of the stopping times S and T , we have S A t n n n that X T _^b<XT and X g _^a>Xg 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 <t) = (X -X„ )1(T <t) n 1 b n n n - (X T - b ) l ( X T _^b<XT ) l ( T n < t ) n n n - (a-X„ )1(X_ >a>X_ )1(S <t) b b — b n n n n + (a-X Q )1(X„ >a>Xc )l(S^<t<T ) b b - S n n n n n - 10 ,s If n = 1, then one must substract (a - X Q ) 1 ( X 0 < a) from the right-hand side i n order to get a correct expression for (b-a)l(T 1<t) . We note that f o r a l l n , (X T - X s ) l ( V t ) = X T A t - X s A t - ( X t - X s ) K S n . t < T n ) n n n n n i ( ((s ,T n )dx - (x - x . ) i ( r r s r ) ) ) n n b t S n n n and therefore, i n view of (1), by summing (b—a)l(T ^t) over n > 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 _<b<X )1(T <t) n>l n n n n - I (a-X )1(X >a>X„ )1(S <t) n>l S n S n " S n n - (a-X 0)l(X 0<a) To f i n i s h the proof, we need to examine some of the terms of (2) more c l o s e l y . By d e f i n i t i o n , D +(I) n (X _ < a} = cf> 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)) + s<t l ( J ( a ) ) d B (a) , where the process B(a) i s continuous, of - 11 -support Included i n {X _ = a} . Now, note that s J(a) = {(u),s) e D +(I) : there e x i s t s an n f o r which s = S (to) , and X (w) = a} n s _ n u {(co.s) e D+(I) , X = a < X (u))} ; s s t h i s shows that f c the sections of J(a) are countable; therefore, the i n t e g r a l l ( J ( a ) ) d B (a) i s zero. F i n a l l y , because the set 0 {((0,s) : s = Tn(uO and X T (u) = a) i s contained i n D"(I) , n _ we have that: I (X - a ) l ( D + ( I ) ) l ( X = a < X ) = £ (X - a ) l ( D - ( I ) ) l ( X _=a<X ) s<t fa s<t s s - I (X - a ) l ( X =a)l(T <t) n>l 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_=a<X ) s<t - I ( X T - a ) l ( X T _=a)l.CT n<t) n>l n n + I (X -a)l(X„ =a)lCS <t) n>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 <t) o b - b n n>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 <t) T b b n n>l n n n and (5) I (X T - b ) l ( X T _<b<XT ) l ( T n < t ) n>l n n n = I (X T -b)l(a<X T _<b<XT ) l ( T n < t ) n>l n n n = I (X T - a ) l ( X T _<a<b<XT ) l ( T n < t ) n>l n n n + (a-b) I 1(X T _^a<b<X )1(T <t) n>l n n Noting that Ca-XQ) - (a-X 0)l(X Q<a) = - ( X 0 - a ) + , and su b s t i t u t i n g (3), (4) and (5) back into (.2) f i n i s h e s the proof. QED. Remark: Def i n i t i o n (1,2.1) and Proposition (11.1,1) give r i s e to an expression f o r the differe n c e between (b-a) times the number of upcrossings of the i n t e r v a l I = [a,b] and the l o c a l time at a . Namely, (11,1.2) t l T a (b^a)N+(0,t,a,b) - ± L a = l ( u + ( I ) ) l ( X s _ > 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 _<a<b<XT ) l ( y t ) n>l n n + C ._ (X - a ) l ( X <a<X ) - _ (X - a ) l ( X _ <a<b<XT )1(T <t) ] + C I (X - a ) l ( X =a<X ) - I (X - a ) l ( D - ( I ) ) l ( X _=a<X )] s<t s s g ^ t s s s + C I (a-X )1(X >a>X ) - I (a-X_ )1(X >a>X )1(S <t)] s<t S S S n>l S n S n " S n n - I (X T -b)l(a<X T _<b<XT ) l ( T n < t ) n>l 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-e<X _<x+e) |d(V+J) | = 0 s s f o r a l l but couiitably many x e TR. , a.s. (b) If X i s continuous, and i f t i s such that x -»• L i s continuous, then t l i m sup e+0 xeR. l(u +([x,x+£ ]))1(X >x)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-e<X <x+e)|d(V+J) | < s s o The conclusion now follows from the f a c t that i f i(x s_=x)|dv s| + I |AXS|KXS-=X) s^t - 15 -x £ {y : L y ^ L y } u {X ,X _: s<t, AX #)} , then the right-hand t t s s s side of the i n e q u a l i t y i s zero ( r e c a l l (1.2.3)), and t h i s set i s countable, since both and X are cadlag. Proof of (b)' Suppose that t i s such that 1/ i s continuous. Pick di out of a n u l l set so that f IdV I i s f i n i t e . Note that 0 s by d e f i n i t i o n of U +(«) , l(u +([x,x+ej))l(X s>x) < l(x<X g<x+e) , and therefore, by monotonicity, i t i s enough to show that f (x) •> 0 t n uniformly, where f (x) = J l(x<X <x + l/n)|dV | . C l e a r l y , the n Q s s f^'s are decreasing, and by dominated convergence, ^ n ( x ) ^ 0 point-wise. Now, consider g(x) = J 1 (X < x) | dV | . I f y 4- x , o s s then g(y) -> g(x) by dominated convergence, and i f y '+ x , then t g(y) converges to J l ( X <x)|dV | which i s equal to g(x) since o s s |dVg| does not charge the l e v e l sets {X g = x) by co n t i n u i t y of • L . This shows that f (•) i s continuous for each n , since t n f n ( x ) = g(x + l/n) - g(x) . To f i n i s h the proof, we note that for f i x e d co , the maps r n ( " ) have compact support, so that by Dini's theorem, sup f (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<X <x+e)dV | goes Q s s to zero with e whenever x i s a point of continuity of 1/ . As we have a sharp L P estimate to handle s i m i l a r i n t e g r a l s , i t i s natural to turn to a B o r e l - C a n t e l l i argument to get a uniform bound for the martingale part of the stochastic i n t e g r a l of (II.1.2). F i r s t , as i n [1], we d i s c r e t i z e the state-space by - 16 -defin i n g a c o l l e c t i o n of grids; f o r k e B , l e t R k = Uk 7 : i e 7L , | i | <k8} (II.2.2) LEMMA: Let X be a semi-martingale l o c a l l y i n H 2 Then, f o r any B > 0 , the set t s {max sup xeR, t<B I k 0 l(u + ( [ x , x+k. 6])>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° 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 „n, , -o, Denote by X^ , (X C)° and Y^(x,k b) the processes X_ A , t -ft n x£ and Jl(u +([x,x+k ]))1(X n_>x)d(X°) 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<B t < P{max sup|Y^(x,k 6 ) | > k " 1 / 2 i.6 .} + P{a < B} xeR k t<B n Using Chebyschev's i n e q u a l i t y and the fac t that #R^ = 2k , l e t t i n g c be a constant depending only on n whose value may change from l i n e to l i n e , one obtains P{max sup | Y^(x,k" 6)| > k~ 1 / 2} xeRfc t<B - 17 -< ck 1 0E{sup|Y^(x,k - D ) r} t<b ,11, n -6. I 4-, < c k 1 0 E { ( l(u +([x,x+k 6 ] ) ) l ( x n _ > 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)<l(x<X n_<x+k~^) s s and L P-estimate (1.2.4) to see that the right-hand side i s majorised -2 by ck . Therefore, by the B o r e l - C a n t e l l i lemma, P{max sup|Y^(x,k ^) | > k i.o.} = 0 for any n . Pick n xeR. t<B k large enough so that > 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 —1/2 —6 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<B Note: To prove t h i s , i t i s enough to show that f o r each t , (*) l i m sup | eN +(0,t,x,x+e) - j L X | = 0 a.s. e-K) xe]R Indeed, given any <S > 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„ < t„ < ... -, 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 £ [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+£) - y L X | < | eN+(0,t,x,x+£") - £N +(0,t i,x,x+£) | + |£N +(0, t l,x,x+£) - | L X I + | | L X - | L X | i i < |£N +(0,t 1 + 1,x 1x+£) - 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<t S (3) max |k 6N +(0,t,x,x+k 6 ) - y L X | < <5 , for a l l k> 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<x + e (here, the l a s t i n e q u a l i t y uses the fact that (2) y i e l d s (m + 2)"6 + (m + 2)"7 < (m + l ) ~ 6 .) For convenience, set I . = [x , , x - +(m-1) ] m-I m-1 m-1 a n d 42 = Cxm+2 + ( m + 2>" ?» 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 + °° ; 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 ^ , „ ) - ^ - 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? = » , and l i m = -» . 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<s<t, l i m sup K t(X,P(a,e)) e+0 a s = L ( t ) - L(s) . 2. A s p e c i a l case It was shown i n [1] that the arc length of Brownian motion i s almost surely L(t) = t ; i n f a c t , i t was shown that |K^(B,Q) - ( t - s ) | converges to zero uniformly over a l l p a r t i t i o n s Q e i ( v ) as v 0 . Note that the uniformity of t h i s convergence f a i l s when p a r t i t i o n s of the time axis are considered [7]. The argument used i n the proof s t i l l holds for the semi-martingales of theorem - 22 -'(II.2.4) and y i e l d s the following: (III.2.1) PROPOSITION: Let X be a continuous semi-martingale whose l o c a l time process i s bicontinuous. Then, f o r a.a. sample paths, for a l l B > 0 , li m sup I K ^ X . Q ) - (<X,X> - <X,X> )| = 0 v+0 Qeir(v) s t s t<B Note: By d e f i n i t i o n , K^(X,Q) i s add i t i v e on path segments up to an error of sup |x (U))|'|[Q|| , and therefore, i t i s s u f f i c i e n t to s<t S assume s = 0 . Moreover, an argument s i m i l a r to the one i n the note preceeding the proof of (II.2.4) uses the uniform co n t i n u i t y of <X,X> 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) - <X,X> | = 0 a.s. for each t v-K) Q£TT(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) - <X,X> ! 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-«J)N-(0,t,xJ+1)l ( x ) - ± L X | d x E Q £ 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 , ^ ) ! «-Kl ^ i ^ i + i ) xeJR x Q , sup Kx^ -xXw.t.xJ.x^ )-!^ 1 I + Qeu (v) 1 X - Q i • I 1 T 1 1 T X I s ? p q Q h L t - 2 L t l x€Lx^,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 <X,X> 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. . £+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+£] 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 u<s u u u B (x) = T (x-X )1(X >x>X ) s L' u u - u u<s C (x) = y (X - x ) l ( X =x<X ) s u u~ u u<s For each x e TR , A f c(x) , B t(x) and C f c(x) are f i n i t e . Now f i x an x e TR . By ( I I . 1 . 3 ) , the following holds for any e > 0 : E t(x,x+e)| < |A t(x) - I (X T - x ) l ( X T _<x<x+e<XT ) l ( T n < t ) n>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+«_,))l(X _=x<X ) s<l s s + I (X T-(x+e))l(x<X T _<x+g<XT ) l ( T n < t ) n ^ l n n n + e I 1(X T _<x<x+e<XT HCTSt) n>l 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_ <x<x+e<XT )1(T <t) i;, T T - T n n>l n n n = I (X - x ) l ( X -<x<x+e<X ) = . ^_ s s s 8 " o 1(X >x+e)dA ( x) s s Therefore, |A t(x) - _ - x ) l ( X T _<x<x+e<XT )1(T St)| = n>l n n n 1(X <x+e)dA (x) s s and by dominated convergence and the d e f i n i t i o n of A g(x) , t h i s i n t e g r a l goes to zero as e -> 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-(x<X T _<x+e<XT ) l ( T n < t ) n>l n n . n < J (X -(x+e))l(x<X _<x+e<X ) < t; s s s s<t Q t l(x<X -<x+e)dK(X) and e I K X T ^<x+£<X )1(T <t) n>l n" n n < e I 1(X _<x<x+e<X ) = s<t (X^x" ) 1 ( X s ^ + e ) d (A+c> 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+£ ( 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 s<t = I (x-X )1(X >x+e>x>X_ )1(S <t) i t> t> - b n n>i n n n < I (x-X )1(X >x>Xc )1(S £t) n>l S n S n ~ S n n < I (x-X )1(X >x>X ) = B. ( x ) s<t S S s fc Therefore, |B (x) - £ ( X - X _ )1(X Q >x>Xo )1(S <t)I < t n>l Sn S - " S n n n KX s_<x+e)dB s(x) and by d e f i n i t i o n of B (x) , t h i s i n t e g r a l goes to zero as s £ + 0 . ( i i i ) F i n a l l y , using the d e f i n i t i o n of D~(») , one finds that t C (x) - I (X -x)l(D-(Cx,x+eJ))l(X _=x<X ) = s<t l(u-(Cx,x+e]))dC (x) 0 But i f s e u ([x,x+e]) , then X <x + £ , and so t h i s l a s t i n t e g r a l - 27 -t i s majorized by / l ( X <x+e)dC (x) which goes to zero as e -»- 0 o s S This completes the proof of (a). Proof of (b) : F i x t > 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 £ > 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 <b)dA (a) < Jl(X-€<X <x+e)dK(X) , s Q s s Q s s and therefore, i t follows from the ca l c u l a t i o n s of ( i ) that: (1) |A t:Ca)- I (X T - a ) l ( X T <a<b<XT ) l ( T n < t ) | < n>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-e<X s<x+e)dK(X) (2) I -b)l(a<X T _<b<XT ) l ( T n < t ) < n>l n n n l(x - e<X <x+e)dK(X) s s 0 Moreover, using simple properties, one obtains (b-a) I 1(X _<a<b<X ) s<t < (b-a) I 1(X _<x-£<a<b<x+e<X )+ _ (AX )l(x - e<X _<a<b<x+e<X ) s<t S S s<t S S S + I (AX s)l(x-€<X s_^a<b<X s<x+e)+ _ (AX s)l(X s_<x-e<a<b<X g<x+6) and noting that the sum (b-a) _ 1(X _^x-e<x+e<X ) i s majorized s<t t 2 b y / ( T T T ^ T T ) 1 ^ -^x+e)l(X >x+e)d(A+C) (x) shows that: s (3) (b-a) I KX^ _<a<b<XT ) l ( T n < t ) n>l n n - 28 -t t ( x _( x- e )) 1(X sSx+€)d(A+C) (x)+ 0 l(x-e<X <x+e)dK(X) s s + 2 l(x-e<X _<x+e)dK(X) s s Now, i t follows from the d e f i n i t i o n of B g(a) and the c a l c u l a t i o n i n ( i i ) that (4) |B (a) - I (a-X )1(X >a>Xr ) 1 ( S <t) | < n>l b n b-~ s-' n n ~n 1 (x-e<Xg_<x+e)dK(X) And f i n a l l y , the d e f i n i t i o n of C g(x) and ( i i i ) y i e l d : l(x-e<X <x+e)dK(X) s s (5) |C (a)- I (X -a)l(D-([a,b]))lCX =a<X )|< s<t S s s C I n e q u a l i t i e s (1) through (5) show that: l i m ^ sup j E t ( a , b ) | ^ 4 I | AX g| l(X g_=x) + 3 I |AXJI(XC=X) €•^-0 [a,b3=[x-e,x+e] s<t + l i m ^X.(x-£))1(VX+£)d(A+C)s(x> s<t s' s where the right-hand side i s zero by choice of x and dominated convergene- convergence. This completes the proof of the lemma. QED. As t h i s lemma indicates, the convergence of E t(a,b) as a t x and b 4- x f a i l s around points i n the range of d i s -c o n t i n u i t i e s of X ; t h i s does not hinder our chances of generalizing (III.2.1), so long as we are able to show that the convergence of (b-a)N +(0,t,a,b) to L X i s uniform over a l l small i n t e r v a l s - 29 -[a,b] not containing these points. Before proving t h i s g e n e r a l i z a t i o n , we present a lemma which 2 w i l l enable us to remove an H hypothesis on X : (III.3.2) LEMMA: Let X be a semi-martingale such that K(X) t i s a.s. f i n i t e f o r each t . Define the process Y = Arctg(X t) . Then Y i s a semi-martingale l o c a l l y i n H P f o r any X X p > 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(«) 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(«) 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 d<Y C, Y C >g = (l4X 2_)~ 2d<X C,X C> s ; therefore, using the f a c t that c c Arctg(«) i s one-to-one and the f a c t that d<Y , Y > g does not charge the sets {s<t : X g / Xg_} , one obtains that f o r each x e TR and e > 0 , (1+x 2) 2Ce/A(x,e))e 1 t l(x<X <x+e.)d<XC,Xc> s s ^ (A(x,e)) 1 0 t 1 (Arctg (x) <Y g<Arctg (x+e ) ) d<YC, Y C> g J 0 t > (l+(x+ e) 2) 2(£/A(x,e)) e 1 J 0 l(x<X <x+e)d<XC,XC> 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 | <v . (III.3.3) THEOREM: Let X be a semi-martingale such that K(X) f c i s a.s. f i n i t e f o r each t . Then with p r o b a b i l i t y one, (a) l i m sup | |l|N +(0,t,I) - j L x | = 0 \H-0 IeS(x,v) t<B f o r any B > 0 , at a l l but countably many x e ]R . (b) l i m sup |Kfc(X,Q) - (<X C,X°> ^ <X°,X C> )| = 0 v+0 Qeir(v) s t s t<B for any B > 0 . c c In p a r t i c u l a r , f o r a.a. sample paths, <X ,X > 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,«,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) - <XC,XC> | = 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<t, AX (co) 4 0}' and D„(co) = {x : LX(<o) + LX~(co)} By lemma (II.2.1)(a) and lemma (III.3.1)(b), and, since we can 2 assume that X i s l o c a l l y i n H , by lemma (11.2,2), we can pick t co out of a n u l l set so that l i m /l(x-e<X _<x+e) | d(V+J) |=0 f o r e+0 0 S S' a l l x | D 1 u D 2, lim sup |-E (a,b)| = 0 f o r a l l e->-0 [a,b]cCx-e ,x+e] x \ D and max | J l (u+([x,x+k~ 6])) 1(X _>x)dM |<k~1/'2 for 1 xeR k0 S s large enough k . Pick x | u . By r i g h t - c o n t i n u i t y of l o c a l time and the choice of co above, there e x i s t s e = e (co) , N = N(co) and B = B(co) >T such that the following hold: - 32 -(1) 1(x-e<X g_<x+e)|d(V+J) | < <5 (2) sup |E tCa,b)|<6 [a,b]c[x - e,x+ e ] (3) max | l(u +(Cx,x+k 6 ] ) ) 1 ( X _>x)dM | < k 1 / 2 for a l l k > N X £ R k O (4) I"J-^ t ~ " i ^ t ^ < ^ whenever | y-x | < £ (5) sup |X (o))|+KN (6) s<t t dV I + K(XV + sup L X < B s t t (7) max(N 1 / 2,(1+2N 1)6-1) < 6/B (8) N 6 < e/8 (9) (k-1) 7 - (k-1) 6 > 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 —fi 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-£/8, x + £ / 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 £ : 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,») 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° ) - ^ L X|} l£S(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-\£ \ + ( 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—1 1 x sup | — 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)£ 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 — L < B , sup | (m+l)" 6N +(0,t,I ) - ^ L X | < H5 l€S(x,v) m 1 t An i d e n t i c a l argument show that: sup |m"V(0,t,I°._) - ^ 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: » |KQ(X,Q) - <X C,X°>: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 ) € 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^,„ 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<B In t h i s new conclusion, i t i s cl e a r that the assumption of b i c o n t i -nuity of l o c a l time i s a necessary one: consider r e f l e c t i n g Brownian motion, l e t L t be i t s l o c a l time process and consider the i n t e r v a l s 1 (x) = Cx-- , x+-] . Then, N +(0,t,I (0)) = 0 for a l l n,t, n n n n and (2/n)N +(0,t,I (0)) - (1/2)L^ doesn't converge to zero, so n t unless l o c a l time i s continuous i n the state-space v a r i a b l e , the I I 4- X I^lN ( 0 , t , I n ) to L f c , along an a r b i t r a r y sequence of i n t e r v a l s I containing the point x f a i l s . S t i l l , f o r r e f l e c t i n g (or skew) Brownian motion, an almost sure point-wise r e s u l t e x i s t s for the convergence of + x 2 e N (0,t,x,x+e) to . Consider a semi-martingale X such that K(X)^_ i s a.s. f i n i t e f o r each t whose l o c a l time process has fixed d i s c o n t i n u i t i e s i n the state-space v a r i a b l e . Then, because the jumps, of L^ . aren't random, (see [8] for the skew Brownian motion case) we can use the point-wise r e s u l t of E l Karoui i n [4] to c o l l e c t n u l l sets where the convergence f a i l s at a l l the (countably many) jump points before carrying on with the proof of (III.3.3)(a). This y i e l d s that f o r such processes, outside a singl e n u l l set, e 1^(0,t,x,x+e) converges to (1/2)L X for a l l x . This i s as far as the techniques used i n t h i s study go, however, and the question of almost sure point-wise convergence - 37 -(when the underlying process i s a continuous semi-martingale) remains open. There i s one l a s t remark to make on theorem (III.3.3). Part c c (b) , besides shedding new l i g h t on the nature of <X ,X > , 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<XC,Xc> = /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^ » 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£es des semi-martingales (CAS continu ), Asterisque, Vol. 52-53 (1977), pp. 63-72. [4] E l Karoui, N. Sur l e s mont£es 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£s A certaines semi-martingales, Asterisque, Vol. 52-53 (1977), pp. 23-36. 

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