SOME LIMIT THEOREMS CONNECTED WITH BROWNIAN LOCAL TIME

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1 SOME LIMIT THEOEMS CONNECTED WITH BOWNIAN LOCAL TIME AOUF GHOMASNI eceived 26 October 24; evised April 25; Accepted 2 April 25 Let B = (B t ) t be a standard Brownian motion and let (L x t ; t, x ) be a continuous version of its local time process. We show that the following limit lim (/2ε) t {F(s,B s ε) F(s,B s + ε)}ds is well defined for a large class of functions F(t,x), and moreover we connect it with the integration with respect to local time L x t.wegiveanillustrativeexample of the nonlinearity of the integration with respect to local time in the random case. Copyright 26 aouf Ghomrasni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.. Introduction.. ThelocaltimeoftheBrownianmotionB at the point a is defined as follows: L a t t = Plim ( Bs a ε) ds, (.) 2ε which equivalently could be written as follows: L a t = Plim 2ε Here we are, more generally, interested in the limit in L : lim 2ε t t ( ) (Bs ε a) (Bs+ε a) ds. (.2) { F ( s,bs ε ) F ( s,b s + ε )} ds (.3) for some function F. Our motivation comes from the desire to connect Chitashvili and Mania results [] with those of Eisenbaum [2]. Applied Mathematics and Stochastic Analysis Volume 26, Article ID 2696, Pages 5 DOI.55/JAMSA/26/2696

2 2 Some limit theorems connected with Brownian local time.2. We give an example which illustrates that the integration with respect to (L x t ; t, x ) does not admit a linear extension in the random case (see Section 3.2 for details) and in particular local time is not a -integrator, which is also proved by Eisenbaum [2]. 2. Notation and preliminaries Let B = (B t ) t be a standard Brownian motion and let (L x t ; t, x ) be a continuous version of its local time process. Let ( t ) t denote the natural filtration generated by B. Without loss of generality, we restrict our attention to functions defined on [,]. For a measurable function f from [,] into, define the norm by ( f =2 f 2 (s,x)e x2 /2s dsdx ) /2 + xf(s,x) e x2 /2s dsdx 2πs s 2πs. (2.) Let be the set of functions f such that f <. In Eisenbaum [2], it is shown that the integration with respect to L is possible in the following sense. Let f Δ be an elementary function on [,], meaning that f Δ (t,x) = (s i,x j) Δ f i,j (si,s i+](t) (xj,x j+](x), (2.2) where Δ ={(s i,x j ), i n, j m} is an [,] grid, and, for every (i, j), f ij is in. For such a function, integration with respect to L is defined by f Δ (s,x)dl x s = ( ) x j+ f i,j L s i+ L xj+ s i L xj s i+ + L xj s i. (2.3) (s i,x j) Δ Let f be an element of. For any sequence of elementary functions ( f Δk ) k N converging to f in, the sequence ( f Δk (s,x)dl x s ) k N converges in L. The limit obtained does not depend on the choice of the sequence ( f Δk ) and represents the integral f (s,x)dl x s. Theorem 2. (see [2]). Let (A(x,t); x, t ) be a continuous random process taking values in, such that for any t in [,] and any ω, A(,t) is absolutely continuous with respect to dx. Note A/ x its derivative and ask A/ x to be continuous. Then A(x,s)dL x s exists and the following hold: (i) for any couple (a,b) in 2 with a<b t a t A(x,s)dL x s = A ( Bs,s ) t t ds+ A(b,s)d s L b s b x A(a,s)d s L a s ; (2.4) (ii) A(x,s)dL x s = A ( Bs,s ) ds; (2.5) x (iii) ( t a ) t a A(x,s)dL x s (ω) = A(x,s)(ω)dL x s (ω). (2.6) b b

3 3. Main results 3.. Deterministic case aouf Ghomrasni 3 Theorem 3.. Let F be a bounded element of. The following equalities hold in L : t { ( ) ( lim F s,bs F s,bs ε )} t ds = F(s,x)dL x s ; (3.) ε t { ( lim F s,bs + ε ) F ( )} t s,b s ds = F(s,x)dL x s ; (3.2) ε t { ( lim F s,bs ε ) F ( s,b s + ε )} t ds = F(s,x)dL x s. (3.3) 2ε emark 3.2. () If we take F(t,x) = (x a) in (3.), we have the very definition of L a t. (2) Eisenbaum [2]has shown that for any Borelian function b(t), t t lim ( Bs b(s) <ε)ds = (,b(s)) (x)dl x s in L, (3.4) 2ε which corresponds to (3.3)withF(t,x) = (x b(t)). Proof. Define H ε (t,x) = (/ε) x x ε F(t, y)dy.thenh ε F in as ε. On the one hand, ( / x)h ε (t,x) = (/ε){f(t,x) F(t,x ε)}. It follows that (see Eisenbaum [2, Theorem 5.(ii)]) t H ε (s,x)dl x s = (/ε) t {F(s,B s) F(s,B s ε)}ds. On the other hand, t H ε (s, x)dl x s t F(s,x)dL x s in L. Corollary 3.3 (see [3]). The following relation holds in L : t t lim g(s)i(b(s) ε<b s <b(s)+ε)ds = g(s)dl b s (3.5) 2ε for a continuous function g :[,t] and a continuous curve b( ) with bounded variation on [,t]. Proof. We apply Theorem 3. to the function F(t,x) = g(t)i(x<b(t)). It follows that (/ 2ε) t g(s)i(b(s) ε<b s <b(s)+ε)ds t g(s)i(x<b(s))dl x s in L as ε. We conclude using (see [4, Corollary 2.9]) that for the continuous function g, wehave t g(s) s Ls b(s) = t g(s)dl b s andom function case. Let a,b be in with a<b.let be the set of elementary processes A such that A(s,x) = (s i,x j) Δ A ij si,s i+](s) (xj,x j+](x), (3.6) where (s i ) i n is a subdivision of (,], (x j ) j m is a finite sequence of real numbers in (a,b], Δ ={(s i,x j ), i n, j m}, and,isa ij an sj -measurable random variable such that A ij forevery(i, j).

4 4 Some limit theorems connected with Brownian local time Eisenbaum [2] asked the following question: does integration with respect to (L x t ; t, x )admitalinear extension to the field generated by, verifying the following property? If (A n ) n converges a.e. to A(t,x), then ( b a A n (s,x)dl x s ) n converges in L to b a A(s,x)dL x s. She only obtained a negative answer to the following weaker question: { b } Is the set A(s,x)dL x s, A bounded in L? (3.7) a Consequently, integration with respect to (L x t ; t, x ) does not admit a continuous extension in L. Here we give an illustrative example, thanks to a result obtained by Walsh, which shows the lack of a linear extension. Let us define A ε (t,x) = (/ε) x x ε L y t dy and à ε (t,x) = (/ε) x+ε x L y t dy. We see easily that A ε (t,x)(resp.,ã ε (t,x)) converges a.e. to L x t, nevertheless we have lim t emark 3.4. The integrals t A ε (s,x)dl x s lim t A ε (s,x)dl x s and t Theorem 2., however, one does not know whether t à ε (s,x)dl x s. (3.8) Ãε(s,x)dL x s are well defined thanks to L x s dl x s is well defined or not. Let us recall, for the convenience of the reader, Walsh s theorem about the decomposition of A(t,B t ):= t {Bs Bt}ds. Theorem 3.5 (see [5]). A(t,B t ) has the decomposition where X t = lim 2ε t { L B s s A ( t,b t ) = t L Bs s db s + X t, (3.9) L Bs ε } s ds = t +lim 2ε The limits exist in probability, uniformly for t in compact sets. t { L B s+ε s } L Bs s ds. (3.) Our example follows by recalling the following property: t t A ε (s,x)dl x s = { L B s s L Bs ε } s ds. (3.) ε eferences []. J. Chitashvili and M. G. Mania, Decomposition of the maximum of semimartingales and generalized Itô s formula, New Trends in Probability and Statistics, Vol. (Bakuriani, 99), VSP, Utrecht, 99, pp [2] N. Eisenbaum, Integration with respect to local time, Potential Analysis 3 (2), no. 4,

5 aouf Ghomrasni 5 [3] G. Peskir, A change-of-variable formula with local time on curves, esearch eport 428, Department of Theoretical Statistics, University of Aarhus, Aarhus, 22, 7 pp. [4] Ph. Protter and J. San Martín, General change of variable formulas for semimartingales in one and finite dimensions, Probability Theory and elated Fields 97 (993), no. 3, [5] J.B.Walsh,Some remarks on A(t,B t ), Séminaire de Probabilités 27, Lecture Notes in Mathematics, vol. 557, Springer, Berlin, 993, pp aouf Ghomrasni: Fakultät II Mathematik und Naturwissenschaften, Institut für Mathematik, Technische Universität Berlin, Straße des 7. Juni 36, 623 Berlin, Germany address: ghomrasni@math.tu-berlin.de

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