On It^o's formula for multidimensional Brownian motion

Transcription

1 On It^o's formula for multidimensional Brownian motion by Hans Follmer and Philip Protter Institut fur Mathemati Humboldt-Universitat D-99 Berlin Mathematics and Statistics Department Purdue University West Lafayette IN , USA Abstract. Consider a d-dimensional Brownian motion = ( ; : : : ; d and a function F which belongs locally to the Sobolev space W ;2. We prove an extension of It^o's formula where the usual second order terms are replaced by the quadratic covariations [f (; ] involving the wea rst partial derivatives f of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(; ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and bacward stochastic integrals. Key words: It^o's formula, Brownian motion, stochastic integrals, quadratic covariation, Dirichlet spaces, polar sets. Supported in part by ONR grant # N and NSF grant # 949-INT

2 . Introduction The behavior of a smooth function F on R d along the paths of d-dimensional Brownian motion is described as follows by It^o's formula. Let P x be the distribution of Brownian motion with initial point x, and let = ( ; : : : ; d denote the coordinate process on the canonical path space = C([; ; R d. Consider the process A dened by (: A t = F ( t? F (? d = t f ( s d s ; where we denote by the partial derivatives of F. It^o's formula provides an alternative description of the process A: (:2 A t = 2 t F ( s ds P x? a:s: for any t, and for any starting point x 2 R d. Note, however, that the description (.2 in terms of the Laplace operator involves second order dierentiability of F, while denition (. requires only dierentiability of rst order. In fact, the process in (. is well dened whenever F belongs to the Sobolev space W ;2, at least locally. In this case, we choose an appropriate version of F and use the wea rst derivatives f in order to dene (. P x - almost surely for all x =2 E, where E is some polar set. Thus the question arises how to formulate an analogue to (.2 for a general function F 2 W ;2 loc. Of course we can always approximate F by smooth functions F (n in such a way that the terms in (. converge to the corresponding terms for F, and then we get the description (:3 A t = lim n! 2 t F (n ( s ds: But rather we are interested in an intrinsic description which directly involves the function F itself. It turns out that such an intrinsic description can be given in terms of quadratic covariation. We show that for any initial point x 2 R d, except for some polar set, the quadratic covariations [f (; ] exist as limits in probability of the usual sums under the measure P x. Our extension of It^o's formula consists in identifying the process A dened by (. as (:4 A t = 2 for all x except for some polar set. d = [f (; ] t P x? a:s: 2

3 If F is the dierence of two positive superharmonic functions so that the distribution F is given by a signed measure, then (.5 provides an explicit description of the 2 additive functional associated to which appears in the extended It^o formula of Brosamler (97 and Meyer (978. In the general case F 2 W ;2, we can view F as a function in the Dirichlet space associated to d-dimensional Brownian motion. From this point of view, A is the process of zero energy appearing in Fuushima's decomposition of the process F ( t (t ; cf. Fuushima (98. Thus, our formula (.5 provides an explicit construction of the process of zero energy in terms of quadratic covariation. In the one-dimensional case, the extension (.5 of It^o's formula was shown in Follmer, Protter and Shiryaev (995. In this paper we consider the case d 2. The basic idea is the same: The existence of the quadratic covariations in (.4 is shown by proving that the forward and the bacward stochastic integrals of f ( can be approximated by the corresponding sums. But in contrast to the one-dimensional case, these approximation results hold only for all initial points x outside some exceptional set of capacity zero, and the proofs are more subtle. In section 2 we x a starting point x 2 R d and a measurable function f on R d. We formulate two integrability conditions on f in terms of x which guarantee that both the forward and the bacward stochastic integral can be constructed in a straightforward manner as limits in probability (:5 t f( s d s = lim n! <t i <t f( ti ( t i+? t i and (:6 t f( s d s = lim n! <t i <t f( ti+ ( t i+? t i under the measure P x. This implies the existence of the quadratic covariations (:7 [f(; ] t = lim n! <t i <t ff( ti+? f( ti g( t i+? t i as limits in probability under the measure P x, and their identication as dierences (:8 [f(; ] t = t f( s d s? t f( s d s of bacward and forward stochastic integrals. In section 3 we consider a measurable function f such that (:9 P x [ t f 2 ( s ds < ] = : 3

4 at least for some x and for some t. Note that condition (.9 is clearly a minimal requirement if we want to tal about stochastic integrals of f(. Using results of Hohnle and Sturm (993 on multidimensional analogues of the Engelbert-Schmidt - law, we show that condition (.9 implies that our integrability conditions in section 2 for the existence of the quadratic covariations [f(; ] are satised for all starting points except for some polar set. In section 4 we apply these results to the wea derivatives f of a function F 2 W ;2. This leads us to our characterization (.5 of the process A dened by (.. Using the identication (.8 of the quadratic covariations [f (; ], our version (.5 of It^o's formula can also be written in the form (: F ( t? F ( = d = t f ( s d s ; where for a function f 2 L 2 loc (Rd we dene the Stratonovich integral as (: t f( s ds = t 2 ( f( s ds + t f( s d s : The idea of deriving an extended It^o formula in terms of quadratic covariations dened by (.8 or in terms of Stratonovich integrals dened as in (. has appeared independently in Russo and Vallois (996 in a general semimartingale context, and in Lyons and hang (994 in the context of Dirichlet spaces. Note that it maes sense to use both (.8 and (. as a denition of the quantities appearing on the left hand side whenever the processes are semimartingales after time reversal. However, the explicit approximation of the stochastic integrals in (.5 and (.6 and the resulting identication of the quadratic covariations as limits in probability of the sums in (.7 is another matter. Such an approximation is of course straightforward if f is continuous. Russo and Vallois (996 consider a dierent approximation where they rst smoothe the right hand side of (.7 by taing integrals over time instead of the usual sums. In Lyons and hang (994, the identication (.7 of the quadratic covariations [f(; ] is shown under the regularity assumption that the function f belongs to the Dirichlet space, and convergence in probability is formulated with respect to a reversible reference measure. In this paper, we concentrate on the classical case of Brownian motion. But here we insist on two improvements. First, the approximations (.5, (.6 and (.7 are established with respect to a given starting point x 2 R d under explicit integrability conditions involving f and x. The second point is that we remove any smoothing and any regularity assumptions on the measurable function f. We require only the minimal integrability conditions which are needed in order to guarantee existence of the forward stochastic integral in (.. Thus, the existence of the quadratic covariations in (.4 is established on exactly the same level of generality which is appropriate for dening the stochastic integrals in (.. 4

5 2. Existence of Quadratic Covariation Let f be a measurable function on R d where d 2. Our purpose in this section is to establish the existence of the quadratic covariations [f(; ] under appropriate integrability hypotheses on f, but without assuming any regularity conditions. Consider the sums (2: <t i <t ff( ti+? f( ti g( t i +? t i along a sequence of partitions D n of R +. As in Follmer, Protter and Shiryaev (995, the idea is to decompose (2. and to show that the two sums and <t i <t <t i <t f( ti ( t i+? t i f( ti +( t i +? t i converge separately, to respectively a forward and a bacward stochastic integral. To this end we assume that the sequence of partitions satises the following conditions: (2:2 lim n! sup t i 2D n (t i+? t i = ; M := sup n t i+ sup < ; t i 2D n t i note that the second condition is satised whenever the partitions are equidistant. For a given point x 2 R d we dene two norms for f: (2:3 jjfjj (x = jf(yj jjx? yjj?d dy and (2:4 jjfjj 2 (x = ( f(y 2 v(jjx? yjjdy 2 ; where (2:5 v(r = (? log r _ if d = 2 r 2?d if d 3 (2.6 Remar. Suppose that f has compact support. If f is also bounded then both norms jjfjj i (x (i = ; 2 are clearly nite for every point x 2 R d. This is still true if f is 5

6 in L p for some p > d; see remar (3.24. In section 3 we will see that, in view of a general result on the existence of quadratic covariation, it is natural to assume niteness of both norms for all points x =2 E, where E is an exceptional set which is not hit by Brownian motion. (2.7 Proposition. Let f be a measurable function on R d with compact support, and let x 2 R d be such that jjfjj 2 (x <. Then the forward stochastic integral satises (2:8 t for each 2 f; : : : ; dg. f( s d s = lim n! <t i <t Proof. It suces to consider only the case t =. Dene the processes and n by (2:9 (!; s = f( s (!; (2: n (!; s = The convergence in (2.8 is equivalent to f( ti ( t i+? t i in L 2 (P x t i 2D n f( ti (!I (ti ;t i+ ](s: (2: lim n! jj? njj 2 = ; where we use the norm (2:2 jj jj 2 = E x [ (!; s 2 ds] 2 for any measurable function on [; ]. Observe that if f 2 C b (R d, then (2. holds by Lebesgue's dominated convergence theorem. The general case will follow by approximating f by continuous functions in the norm jj jj 2 (x : 2 Note that the Gaussian density satises the inequality p s (z = (2s? d 2 exp(?jjzjj 2 =2s (2:3 p s (zds c(rv(jjzjj 6

7 for any z 2 R d with jjzjj R, where c(r is some constant depending on R; see, e.g., Dynin (965, VIII, 8.6. Denoting by K the compact support of f and choosing R sup y2k jjy? x jj, we obtain the estimate (2:4 hence jjjj 2 2 = c(r f 2 (yp s (y? x dyds f 2 (yv(jjy? x jjdy; (2:5 jjjj 2 a 2 jjfjj 2 (x ; where a 2 = p c(r. 3 In order to obtain a similar estimate for the approximating process n, note that (2:6 p ti (z (2t i? d 2 exp(?jjzjj 2 =2s M d 2 ps (z; for t i s t i+, due to our assumption (2.2. Again using (2.3 we get jj n jj 2 = 2 f 2 (y p ti (y? x ((t i+ ^? t i dy (2:7 hence M d 2 <t i <t f 2 (y M d 2 c(rjjfjj 2 2(x ; p s (y? x dsdy (2:8 jj n jj 2 b 2 jjfjj 2 (x where b 2 = p c(rm d 4. 4 Next we choose a continuous function g with compact support such that jjg?fjj 2 (x ", and denote by and n the processes associated to g as in (2.9 and (2.. We have (2:9 jj? n jj 2 jj? jj 2 + jj? n jj 2 + jj n? n jj 2 : But (2:2 lim n jj? n jj 2 = since g is bounded and continuous, and also (2:2 jj n? n jj 2 = jj(? n jj 2 b 2 jjf? gjj 2 (x ; 7 :

8 due to our estimate (2.8 applied to the function f? g. This together with (2.5 implies (2:22 lim sup n! jj? n jj 2 jj? jj 2 + lim sup jj n? n jj 2 n! a 2 jjf? gjj 2 (x + b 2 jjf? gjj 2 (x (a 2 + b 2 ": Since " > was arbitrary we have shown (2. and hence (2.8. (2.23 Proposition. Let f be a measurable function on R d with compact support, and let x 2 R d be such that jjfjj i (x < for i = ; 2. Then the bacward stochastic integral satises (2:24 t f( s d s = lim n! <t i <t f( ti+ ( t i+? t i in L (P x for each 2 f; : : : ; dg. Proof. It suces to consider the case t =. Let P x be the distribution of the time reversed process R under P x, where (R t =?t : The time reversed process R is a d-dimensional Brownian bridge tied down to 2 R d and starting with initial distribution N(; I, where I is the identity matrix. Under P x, each component is a semimartingale with decomposition (2:25 t = + W t + t x? s? s ds; where W ( = ; : : : ; d are independent Wiener processes. The convergence in (2.24 for t = is equivalent to the convergence (2:26 f( s d s = lim n! s i 2D n <s i < f( si ( s i+? s i in L (P x ; where D n = f? t i jt i 2 D n g. Let us now use the decomposition (2.25 of under P x, and let us rst show that condition jjfjj 2 (x < implies (2:27 lim n! s i 2D n <s i < f( si (W s i+? W s i = f( s dw s in L 2 (P x : This follows as in the proof of proposition (2.7. We have only to chec that the estimates (2.5 and (2.8 have analogues in terms of the norm jj jj 2 dened by (2:28 jj jj 2 = E x [ (!; sds] 8

9 for any measurable function on [; ]. This is clear for (2.5 since (2:29 jjjj 2 = 2 E x [ = E x [ f 2 ( s ds] f 2 (? sds] = jjjj 2 2 a 2 jjfjj 2 2(x : In order to obtain an analogue to (2.8, consider the term (2:3 jj n jj 2 2 = E x [ = E x [ = s i 2D n <s i < <t i < f 2 (y f 2 ( si (s i+? s i ] f 2 ( ti+ (t i+? t i ] <t i < For t i s t i+ and for any z 2 R d we have p ti+ (y? x (t i+? t i dy: (2:3 p ti+ (z (2s? d 2 exp(? jjzjj 2 2t i+ (2s? d jjm? 2 zjj 2 2 exp(? = p s (M? 2 z 2s due to (2.2. Using again the estimate (2.3, and observing that v(m? 2 r v(r for some constant which only depends on M and d, we get (2:32 <t i < p ti+ (z(t i+? t i Returning to (2.3 we see that p s (M? 2 zds c(rv(jjm? 2 zjj c(rv(jjzjj: (2:33 jj n jj 2 b 2jjfjj 2 (x for some constant b 2. Using the estimates (2.29 and (2.33, we can now conclude as in part 3 of the proof of proposition (2.7 that (2.27 holds. 2 It remains to show (2:34 lim n! s i 2D n <s i < si+ s f( si? x s i? s ds = f( s s? x? s ds in L (P x 9

10 or, equivalently, (2:35 lim n! <t i < Let us dene the norm (2:36 jj jj = E x [ ti+ s f( ti+? x t i? s ds = f( s s? x? s ds in L (P x : j (!; sj j s? x j ds] s for any measurable function on [; ]. For the process dened in (2.9 we have (2:37 jjjj = E x [ = jf( s j j s? x j ds] s jf(yjjy? x j p s (y? x dsdy: s However for the Gaussian density p s there is a constant a such that (2:38 jjzjj p s (z ds a jjzjj?d s for any z 2 R d ; see, e.g., Dynin (98, VIII, Combining (2.37 and (2.38 yields (2:39 jjjj a jjfjj (x : 3 We also need an estimate of the form (2.39 for the approximating processes (2:4 n(!; s = t i 2D n f( ti+ (!I (ti ;t i+ ](s: We will write P i for P t i 2D n ;<t i <. Then (2:4 jj n jj = i = i = i E x [jf( ti+ j ti+ ti+ t i js? x j ds] s js E x [? xj dsjf( ti+ j] t i s ti+ t i s E x [E x [js? x jj ti+ ]jf( ti+ j]ds: Recall that a normal random variable with law N(m; 2 satises (2:42 E[jj] = m(2( m? + r 2 exp(?m2 2 2 jmj + r 2

11 where denotes the distribution function of N(;. In the conditional expectation appearing in (2.4, the normal random variable = s? x has conditional mean (2:43 m = s t i+ ( t i+? x and conditional variance (2:44 2 = s t i+ (t i+? s (M? s; where we use our assumption (2.2. Thus, (2.42 implies (2:45 E x [j s? x jj t i+ ] A i (s + B i (s where we put (2:46 A i (s = s t i+ jj ti+? x jj; B i (s = for t i s < t i+. Returning to (2.4 we obtain the estimate r 2Ms (2:47 jj n jj E x [ i ti + t i jf( ti+j s (A i (s + B i (sds]: 4 Let us rst consider the term (2:48 E x [ i ti+ t i jf( ti+j s A i sds] = jf(yj jjy? x jj i t i+? t i p ti+ (y? x dy: t i+ Due to (2.3 and (2.38 we obtain the estimate (2:49 jjzjj i s i+? s i p si+ (z jjzjj s i+ cm d 2 jjzjj?d : s p s(m? 2 zds Returning to (2.48 we see that ti+ E x [ (2:5 i t i s jf( t i+ ja i s ds] jf(yjjjy? x jj?d dy = jjfjj (x where = c M d 2 :

12 5 As to the second term on the right hand side of (2.47, we again use (2.3 to obtain the estimate (2:5 ti+ r jf( ti+j 2M ti+ E x [ B i (sds] = jf(yj ps p ti+ (y? x dsdy i t i s i t i r 2M jf(yj p p s (M? 2 (y? x dsdy: s But for any z 2 R d we have (2:52 for some constant c 2, and so (2.5 implies (2:53 E x [ i ti+ jf( ti+ j t i s p s p s (zds c 2 jjzjj?d B i (sds] l = l jjfjj (x q 2 where l = c 2M d 2. 6 Combining (2.47 with (2.5 and (2.53 we see that (2:54 jj njj b jjfjj (x ; jf(yj jjy? x jj?d dy where b = + l. Using the estimates (2.54 and (2.39, we can now conclude as in part 3 of the proof of proposition (2.7 that jj? njj tends to. This implies the convergence in (2.35 and (2.34, and so the proposition is proved. We combine propositions (2.7 and (2.23 to obtain: (2.55 Corollary. Let f be a measurable function on R d, let x 2 R d, and let K m (m be a sequence of compact sets such that (2:56 lim m! P x [ t 2 K m 8 t 2 [; T ]] = 8T > : Assume that for any m the restriction f m of f to K m satises (2:57 jjf m jj i (x < (i = ; 2: Then the quadratic covariation (2:58 [f(; ] t lim n! = t <t i <t f( s d s? ff( ti+? f( ti g( t i+? t i 2 t f( s d s

13 exists in probability under P x and satises t t f( s d s? f( s ds ; (2:59 [f(; ] t = for each 2 f; : : : ; dg. Proof. Let t be xed and " >, and let T m = infft > j t =2 K m g be the exit time from K m. We denote by S n the n-th sum in (2.58, by S the dierence of the forward and bacward stochastic integrals, and by Sn m and S m the corresponding terms if the function f is replaced by f m. Since S n = Sn m and S = Sm P x -a.s. on ft m > tg, we have (2:6 P x [js n? Sj "] P x [T m t] + P x [js m n? S m j "] for any m. Applying propositions (2.7 and (2.23 to the function f m we see that the last term converges to as n tends to. Thus, (2:6 lim sup n! P x [js n? Sj "] P x [T m t]; and due to (2.56 the result follows by letting m tend to. 3. Exceptional sets Let f be a measurable function on R d. In our approximation (2.55 of the quadratic covariation (3: [f(; ] t = t f( s d s? t f( s d s : as a limit in probability under P x, we have assumed integrability conditions on f which are formulated in terms of the initial point x. In this section we show that it is no loss of generality to mae these assumptions for all x outside some exceptional set which is not hit by Brownian motion. (3.2 Denition. A measurable set E R d is called polar if (3:3 P x [ t 2 E for some t > ] = 8x 2 R d : (3.4 Remar. This probabilistic notion of an exceptional set is equivalent to the potential theoretic notion of a set of capacity zero; see, e.g., Fuushima (98, Th and Example Equivalently, we can dene these exceptional sets in terms of the Bessel capacity of order (,2 as in iemer (989, 2.6; see, e.g., Fuushima (993, p.25. 3

14 Note rst that in order to introduce the forward stochastic integral in (3. with respect to the measure P x, at least for some x 2 R d and some t >, we clearly need the condition (3:5 P x [ t f 2 ( s ds < ] = : But the results in Hohnle and Sturm (993, Th.. show that the validity of (3.2 for some x 2 R d and some t > implies that the function f satises condition (3.5 for all t > and for all initial points x =2 E where E is some polar set. Moreover, it follows that all the assumptions we used in theorem (2.55 hold for any starting point lying outside some polar set: (3.6 Proposition. Let f be a measurable function on R d such that condition (3.5 holds for some x 2 R d and some t <. Then there exist a sequence of compact sets K m R d (m and a polar set E such that the conditions (3:7 lim m! P x [ t 2 K m 8 t 2 [; T ]] = 8T > and (3:8 f 2 m (xdx < ; jjf mjj i (x < (i = ; 2 are satised for all x =2 E and for all m, where f m denotes the restriction f I Km of f to the set K m. Proof. It is shown in Hohnle and Sturm (993, Th. 3.5 and 3.7 that the validity of (3.2 for some x 2 R d and some t < is equivalent to the fact that there exists a polar set E 2 and a sequence of compact sets K m (m with (3:9 such that the conditions (3.7 and (3: K m f 2 (xdx < (m K m v( x? y f 2 (ydy < (m are satised for any x =2 E 2. Thus, the restrictions f m of f to K m satisfy (3: f m 2 L 2 (R d ; jjf m jj 2 (x < for all x =2 E 2. 4

15 2 It remains to verify the integrability condition (3:2 jjf m jj (x < for all x outside some polar set. In view of we may assume that f has compact support K and is square integrable. Consider the Bessel potential g jfj of order dened by (3:3 (g jfj(x = g (x? yjfj(ydy; where g is dened as that function whose Fourier transform is (3:4 ^g (x = (2? 2 ( + jjxjj 2? 2 : Note that (3:5 g (z = ( jjzjj?d + o(jjzjj?d with some constant ( as jjzjj! ; see, e.g., iemer (989, p. 65. Thus, there is a constant c such that (3:6 jjy? x jj?d c g (x? y on K and so we have (3:7 jjfjj (x = K jfj(yjjy? x jj?d dy c (g jfj(x : But the function u = g f, being the Bessel potential with index = associated to the square integrable function f, belongs to the Sobolev space W ;2 ; see iemer (989, Th This implies that the version ~u dened by (3:8 ~u(x := lim # vol(b (x B (x u(zdz satises ~u(x < outside some polar set E ; see, e.g., iemer (989, Th.3..4 and 3..2 or Fuushima (993, Th. 2.. Since (3:9 lim # vol(b (x B (x for any y 6= x by Lebesgue's theorem, we obtain (3:2 u(x = lim inf # jfj(y lim vol(b (x # vol(b (x g (y? zdz = g (y? x 5 B (x B (x u(zdz; g (y? xdzdy

16 using Fatou's lemma and Fubini's theorem. For x =2 E, we have thus shown u(x < ; hence jjfjj (x < due to (3.7. Combining proposition (3.6 with corollary (2.55 we obtain: (3.2 Theorem. Let f be a measurable function on R d such that condition (3.5 holds for some x 2 R d and some t <. Then there exists a polar set E such that for any x =2 E the quadratic covariation (3:22 [f(; ] t lim n! = t <t i <t ff( ti+? f( ti g( t i+? t i t f( s d s? f( s ds exists in probability under P x and satises (3:23 [f(; ] t = for each 2 f; : : : ; dg. t t f( s d s? f( s ds ; (3.24 Remar. If f 2 L p loc (Rd for some p > d then the conlusion of the theorem holds for every starting point x 2 R d, without exception. To see this we may assume that f has compact support. In this case, the assumption that f 2 L p (R d for some p > d implies (3:25 jjfjj i (x < (i = ; 2 for every x 2 R d. Thus, our assumptions in corollary (2.55 for the existence of quadratic covariation are satised everywhere. Conditions (3.25 can be veried directly, using Holder's inequality. They also follow from the Sobolev embedding theorem. Note that the function u 2 dened by (3:26 u 2 (x = f 2 (yv( x? y dy belongs to W 2;p=2 ; see Gilbarg and Trudinger (983, Th This implies u 2 2 C(R d for p > d, hence jjfjj 2 (x < for any x 2 R d ; see iemer (989, Th Note also that for f 2 L p the Bessel potential u = g jfj belongs to W ;p ; see iemer (989, Th This implies u 2 C(R d for p > d, again by iemer (989, Th , hence jjfjj (x < for any x 2 R d, due to (3.7. 6

17 4. It^o's formula Let W ;2 denote the Sobolev space of functions in L 2 (R d such that the wea rst partial derivatives belong to L 2 (R d. Recall that a function in W ;2 can be dened everywhere, except for a polar set, in terms of its integral averages, i.e., we can choose a version F and a polar set E such that (4: F (x = lim # vol(b (x B (x F (ydy for all x 2 E ; see, e.g., Fuushima (993, Th. 2. and p.25 or iemer (989, Th Let us now consider a function in W ;2 loc, i.e., a measurable function on Rd which coincides on each compact set with a function in W ;2. We x a version F such that (4. holds outside some polar set E, and we denote by (4:2 2 L 2 loc(r d the -th wea partial derivative of F. With probability, Brownian motion does not enter a given polar set after time, and so the values F ( t (! of the function F along a Brownian path are well dened P x -almost surely for any starting point x =2 E. (4.3 Theorem. Let F 2 W ;2 loc be given as above. For all x 2 R d except for some polar set, the quadratic covariation (4:4 [f (; ] t = lim (f ( ti+? f ( ti (t n i+? t i t i t exists as a limit in probability under P x for each 2 f; : : : ; dg, and It^o's formula holds in the form d t (4:5 F ( t = F ( + f ( s ds + d [f (; ] t P x? a:s: 2 for all t. = Proof. By a localization argument as in the proof of corollary (2.55, we can assume that F has compact support and belongs to W ;2. Since f 2 L 2 (R d, we have (4:6 P x [ t = f 2 ( s ds < ] = for all x except for some polar set; see Fuushima (98, ( Due to (3.2 we can conclude that the quadratic covariations [f (; ] t are well dened as limits in 7

18 probability under P x for all x, except for some polar set. Alternatively we can apply propositions (2.7 and (2.23, using the estimates jjf jj i (x < (i = ; 2 for x =2 E [ E 2 which are implied by (4.2 and (4.25 below. 2 Let x 2 R d. Suppose that we can approximate F by functions F (n 2 C 2 (R d with compact support in such a way that (4:7 F (x = lim n! F (n (x for all x outside some polar set, and that the partial derivatives f F (n satisfy (n (4:8 lim jjf n!? f jj i (x = (i = ; 2 for the two norms introduced in section 2. Due to our estimates (2.5, (2.29, (2.39 we can conclude, as in the proof of (2.7 and (2.23, that (4:9 lim n! and (4: lim n! In particular, t t t f (n ( s ds = f ( s ds in L 2 (P x t f (n ( s d s = f ( s d s in L (P x : (4: [f (; ] t = t f ( s d s? t f ( s d s (n = lim [f n! (; ] t in L (P x : Applying It^o's formula to the functions F (n, we obtain (4:2 2 d = [f (; ] t = lim n! 2 d [f (n = (; ] t = lim n! (F (n ( t? F (n (? = F ( t? F (? d = t d = t f ( s d s f (n ( s d s P x? a:s:; where we have applied once more (4.9 in the last step. 8

19 3 In order to construct such an approximation, let us tae functions F (n 2 C 2 (R d with compact support such that (4:3 n= jjf (n? F jj ;2 < where jj jj ;2 denotes the Sobolev norm in W ;2. In particular, the functions (4:4 h := n= jf (n? f j belong to L 2 (R d since (4:5 ( h 2 (xdx 2 n= ( jf (n? f j 2 dx 2 jjf (n? F jj ;2 < : n= It follows as in part 2 of the proof of proposition (3.6 that the Bessel potential (4:6 (g h (x = g (x? yh (ydy; is nite for all x outside some polar set E. But due to (3.7 we have (4:8 jjh jj (x c (g h (x ; for some constant c, and so we have shown that jjh jj (x < for all x implies (4:9 n= jjf (n? f jj 2 (x < ; =2 E. This by monotone integration, and so we get the desired approximation (n (4:2 lim jjf n!? f jj (x = for all x =2 E. 4 Let us dene (4:2 ~ h := ( n= (f (n? f 2 2 : 9

20 Since (4:22 ~h 2 dx = n= part of the proof of (3.6 shows that (4:23 (f (n? f 2 2 dx < ; K m v(jjx? yjj ~ h 2 (ydy < for all m and for all x except for some polar set E 2, where (K m is a sequence of compact sets satisfying (3.7. In view of the localization argument in the proof of (2.55, we can assume without loss of generality that F vanishes outside K m for some m. Then we get (4:24 n= v(jjx? yjj(f (n? f 2 (ydy < for all x =2 E 2, and this implies the desired approximation (n (4:25 lim jjf n!? f jj 2 (x = for all x =2 E 2. 5 Let us also chec that (4.7 holds for all points x except for some polar set. We have (4:26 (F? F (n (x = lim # vol(b (x B (x (F? F (n (xdx for all points x except for the polar set E involved in the choice of the version F. Since (4:27 n= due to (4.3, we get the existence of (4:28 lim # vol(b (x jf? F (n j 2 W ;2 B (x n= jf? F (n j(xdx < for all x except for a polar set E 3. For x =2 E [ E 3, we can conclude that (4:29 n= jf? F (n j(x n= lim inf # lim inf # vol(b (x vol(b (x B (x B (x n= jf? F (n j(xdx jf? F (n j(xdx < ; and this implies (4.7. Thus, all properties of the approximation which were used in part 2 of the proof are satised for any x =2 E [ E [ E 2 [ E 3. 2

21 References ] Brosamler, G. (97: Quadratic variation of potentials and harmonic functions. Trans. Am. Math. Soc. 49, p [2] Dynin, E.B. (98: Marov Processes vol. I. Springer-Verlag, Berlin-Gottingen- Heidelberg [3] Follmer, H., Protter, Ph. and Shiryaev, A.N. (995: Quadratic covariation and an extension of It^o 's formula. Bernoulli (/2, [4] Fuushima, M. (98: Dirichlet Forms and Marov Processes. Amsterdam: North- Holland [5 Fuushima, M. (993: Two topics related to Dirichlet forms: quasi everywhere convergences and additive functionals. In: Dirichlet Forms. Springer Lecture Notes in Mathematics 563, 2-53 [6] Gilbarg, D. and Trudinger, N.S. (983: Elliptic Partial Dierential Equations of Second Order. Springer-Verlag, New Yor-Berlin-Heidelberg [7] Hohnle, R. and Sturm, K.-Th. (993: A Multidimensional Analogue to the --Law of Engelbert and Schmidt. Stochastics and Stochastics Reports 44, 27-4 [8] It^o, K. (944: Stochastic integral. Proc. Imperial Acad. Toyo, 2, [9] Lyons, T.J. and hang,t.s. (994: Decomposition of Dirichlet processes and its application. Ann. Probability 22, [] Meyer, P.A. (978: La formule d'it^o pour le mouvement Brownian d'apres G. Brosamler. In: Seminaire de Probabilites II. Springer Lecture Notes in Mathematics 649 [] Russo, F. and Vallois, P. (996: It^o formula for C -functions of semimartingales.prob. Theory Relat. Fields 4, 27-4 [2] iemer, W.P. (989: Wealy Dierentiable Functions. Graduate Texts in Mathematics 2, Springer-Verlag, New Yor Berlin Heidelberg 2