On Uniqueness of Solutions for the Stochastic

We study a nonlinear ltering problem in which the signal to be estimated is conditioned by the Wiener process in the observation equation. The main results establish pathwise uniqueness for the unnormalized lter equation and uniqueness in law for the normalized and unnormalized lter equations. 1 Introduction An early work on uniqueness for the stochastic di erential equations of nonlinear ltering is that of Szpirglas [13]. The basic viewpoint adopted in [13] is to regard the measure-valued stochastic di erential equations of nonlinear ltering as entities quite separate from the original nonlinear ltering problem, for which one can formulate the notions of solution (or weak solution), pathwise uniqueness and uniqueness in law, by essentially adapting these concepts from the theory of Itô stochastic di erential equations (for which see Section IV.1 of Ikeda and Watanabe [5] or Section IX.1 of Revuz and Yor [9]). With these notions at hand, it is then established in [13] that pathwise uniqueness and uniqueness in law hold for both the normalized (FujisakiKallianpur-Kunita) and unnormalized (Duncan-Mortensen-Zakai) lter equations, in the case of a nonlinear ltering problem where the signal is a Markov process which is independent of the Wiener process in the observation equation, and the sensor function in the observation equation is uniformly bounded. Our goal is to look at uniqueness for the stochastic di erential equations of nonlinear ltering from a point of view very similar to that of Szpirglas [13], but for a nonlinear ltering problem in which there is dependence of the signal on the Wiener process of the observation equation. In fact, we shall look at the speci c nonlinear ltering problem where the signal fXtg is an Rd-valued process solving an equation of the form dXt = b(Xt) dt +B(Xt) dWt + c(Xt) dVt; (1.1) the Rd1-valued observation process fYtg is de ned by Yt = Wt + Z t 0 h(Xs) ds; (1.2) and f(Wt; Vt)g is a standard Rd1+d2 -valued Wiener process (precise conditions on the mappings b( ), B( ), c( ) and h( ) will be stated in Section 2). The pair (1.1) and (1.2) represents a simple model of a signal and observation in which the signal fXtg depends on the Wiener process fWtg of the observation equation. Motivated by Szpirglas [13], we shall regard the normalized and unnormalized lter equations for this nonlinear ltering problem as measure-valued stochastic di erential equations, de ned quite independently of the ltering problem, and will formulate the notions of weak solution, pathwise uniqueness, and uniqueness in law for the lter equations. Our main result (see Theorem 2.21 to follow) establishes pathwise uniqueness for the unnormalized lter equation, together with uniqueness in law for the unnormalized and normalized lter equations, subject to reasonably general conditions on the mappings b( ), B( ), and c( ) in the signal equation (1.1), and a uniform boundedness condition on the sensor function h( ) in the observation equation (1.2). As will be seen from the discussion of Section 2 (see Remark 2.22) the elegant semigroup ideas used in Szpirglas [13] to establish pathwise uniqueness do not seem to extend to the ltering problem represented by (1.1) and (1.2), where the signal fXtg depends on the observation Wiener process 1 fWtg, and our approach necessarily involves a di erent method of proof. In Section 2 we review the normalized and unnormalized lter equations for the nonlinear ltering problem given by (1.1) and (1.2), de ne weak solutions, pathwise uniqueness, and uniqueness in law for the lter equations, and state the main result, namely Theorem 2.21. We also discuss the relationship of this result with other works on uniqueness for the nonlinear lter equations in Remarks 2.22 and 2.23. Section 3 is devoted to the proof of the main result, while the proofs of various technical facts and lemmas needed in Section 3 are relegated to Section 4 and Section 5. 2 Stochastic Di erential Equations of Nonlinear Filtering Remark 2.1. For easy access we rst summarize most of the basic notation which will be used in the sequel: (i) For a metric space E, let B(E) denote the Borel -algebra on E, let B(E) denote the set of all real-valued uniformly-bounded Borel measurable mappings on E, and, for 2 B(E), de ne the supremum norm by k k := supx2E j (x)j. Likewise, write C(E) for the set of all real-valued continuous mappings on E, and write C(E) for the collection of all members of C(E) which are uniformly bounded. (ii) For a complete separable metric space E, let M+(E) denote the space of all positive bounded measures on the measurable space (E;B(E)), with the usual topology of weak (or narrow) convergence. Then M+(E) is separable and metrically topologically complete, and Exercise 9.5.6 of Ethier and Kurtz [4] shows that a simple variant of the Prohorov metric turns the topological space M+(E) into a complete separable metric space. Also, let P(E) denote the collection of all members ofM+(E) which are probability measures. For 2 M+(E) and a B(E)-measurable and -integrable mapping from E into R, write ( ) or for the integral RE d . (iii) For a vector x in a nite-dimensional Euclidean space Rn, write xk for the k-th scalar entry of x, 8 k = 1; : : : ; n, and write jxj for the Euclidean norm of x, namely jxj2 :=Pnk=1(xk)2. Also, let C1(Rn) denote the set of all in nitely smooth real-valued mappings on Rn, and let C1 c (Rn) be the collection of all members of C1(Rn) with compact support. Finally, let Ĉ(Rn) denote the collection of all members of C(Rn) which vanish at in nity. Now consider a nonlinear ltering problem made up of the following basic elements: E.1 A xed interval of interest [0; T ], with T 2 (0;1). E.2 A complete probability space ( ;F ; P ) carrying a ltration fFt; t 2 [0; T ]g such that F0 includes all null events of ( ;F ; P ). De ned on ( ;F ; P ) is an Rd-valued continuous fFtgadapted process fXt; t 2 [0; T ]g and an Rd1+d2-valued fFtg-Wiener process f(Wt; Vt); t 2 [0; T ]g such that (1.1) holds, where b : Rd ! Rd, B : Rd ! Rd d1, and c : Rd ! Rd d2 are 2 Borel-measurable and locally bounded functions (that is, uniformly bounded over bounded subsets of Rd). E.3 an Rd1-valued observation process fYt; t 2 [0; T ]g de ned by (1.2), where h : Rd ! Rd1 is Borel-measurable, with E " d1 Xk=1 Z T 0 jhk(Xu)j2 du# <1: (2.3) De ne the observation ltration fFY t ; t 2 [0; T ]g by FY t := fYu; u 2 [0; t]g _ N (P ); where N (P ) := fN 2 F : P (N) = 0g: (2.4) From Lemma 1.1 of Kurtz and Ocone [8] there exists some P(Rd)-valued fFY t+g-optional process f t; t 2 [0; T ]g, called the optimal lter, which is de ned on ( ;F ; P ) and satis es t = E[ (Xt)jFY t+] a:s:; 8t 2 [0; T ]; 8 2 B(Rd): (2.5) From (2.3) and Jensen's inequality we see that E " d1 Xk=1 Z T 0 [ u(jhkj)]2 du# <1; and we can therefore de ne the Rd1-valued innovations process fIt; t 2 [0; T ]g by Ik t := Y k t Z t 0 shk ds; 8 t 2 [0; T ]; k = 1; : : : ; d1: (2.6) An important property of the innovations process is that fIt; t 2 [0; T ]g is an Rd1-valued fFY t+gWiener process (see Theorem VI.8.4 of Rogers and Williams [10], observing that the ltration fYtg on p. 322 of [10] corresponds to our fFY t+g). Since fItg is continuous, it is necessarily fFY t g-adapted, thus fIt; t 2 [0; T ]g is a fFY t g-Wiener process. Now de ne m : Rd! Rd (d1+d2) by m(x) := hB(x) c(x)i ; 8 x 2 Rd; and put A (x) := d Xi=1 bi(x)@i (x) + 1 2 d X i;j=1[m(x)mT (x)]ij@i@j (x); 8x 2 Rd; 2 C1(Rd); (2.7a) Bk (x) := d Xi=1 Bik(x)@i (x); 8x 2 Rd; 2 C1(Rd); k = 1; : : : ; d1: (2.7b) 3 For each 2 C1 c (Rd) one easily sees from Itô's formula that the process M t := (Xt) Z t 0 A (Xs) ds; t 2 [0; T ]; (2.8) is a square-integrable fFtg-martingale with hM ;W kit = Z t 0 Bk (Xu) du; t 2 [0; T ]; k = 1; : : : ; d1: (2.9) This observation, together with Theorem VI.8.11 of Rogers and Williams [10], establishes Theorem 2.2. For the nonlinear ltering problem given by E.1, E.2, and E.3, the optimal lter f t; t 2 [0; T ]g satis es t = 0 + Z t 0 s(A ) ds + Z t 0 d1 Xk=1[ s(hk +Bk ) ( shk)( s )] dIk s ; 8 2 C1 c (Rd): (2.10) The relation (2.10) is known variously as the Fujisaki-Kallianpur-Kunita equation, the KushnerStratonovich equation, or the normalized lter equation. Remark 2.3. Since C1 c (Rd) is dense in Ĉ(Rd), with respect to the supremum norm, it must be convergence determining (see Problem 3.11.11 of Ethier and Kurtz [4]). Now it follows from (2.10) that f t; t 2 [0; T ]g is a continuous P(Rd)-valued process, and therefore fFY t g-adapted. Thus, we can replace FY t+ in (2.5) by FY t . The characterization of f tg given by Theorem 2.2 becomes useful when some form of uniqueness is established for (2.10). The approach adopted here is suggested by the work of Szpirglas [13], which in turn is motivated by the results of Yamada and Watanabe [14] on weak solutions, pathwise uniqueness, and uniqueness in law for Itô SDE's (see Section IV.1 of Ikeda and Watanabe [5] or Section IX.1 of Revuz and Yor [9] for a comprehensive account of these ideas). Taking advantage of the fact that the innovations process fItg which \drives" (2.10) is a standard fFY t g-Wiener process, we can follow [13] and regard the normalized lter equation as an entity quite separate from the nonlinear ltering problem, namely as a probability-measure valued stochastic di erential equation driven by a standard Wiener process, for which one can formulate the notions of weak solution, pathwise uniqueness, and uniqueness in law as follows: (compare Szpirglas [13], D e nition III.1, V.1, V.2, and Bhatt, Kallianpur and Karandikar [1], De nition 9.1): De nition 2.4. The pair f(~ ; ~ F ; f ~ Ftg; ~ P ); (~ t; ~ It)g is a weak solution of the normalized lter equation when: 1. (~ ; ~ F ; f ~ Ftg; ~ P ) is a complete ltered probability space; 2. f~ It; t 2 [0; T ]g is an Rd1-valued f ~ Ftg-Wiener process on (~ ; ~ F; ~ P ); 4 3. f~ t; t 2 [0; T ]g is a P(Rd)-valued continuous f ~ Ftg-adapted process such that ~ P Z T 0 d1 Xk=1[~ sjhkj]2 ds <1! = 1; (2.11) and, for each 2 C1 c (Rd), the following holds to within indistinguishability ~ t = ~ 0 + Z t 0 ~ s(A ) ds + d1 Xk=1 Z t 0 [~ s(hk +Bk ) (~ shk)(~ s )] d~ Ik s ; t 2 [0; T ]: (2.12) Remark 2.5. The terminology that (~ ; ~ F; f ~ Ftg; ~ P ) is a \complete ltered probability space" will always be understood to mean that (~ ; ~ F; ~ P ) is a complete probability space carrying the ltration f ~ Ft; t 2 [0; T ]g, and ~ F0 includes all P -null events in ~ F . Remark 2.6. In view of De nition 2.4, it follows that f( ;F ; fFY t g; P ); ( t; It)g for fFY t ; t 2 [0; T ]g, f t; t 2 [0; T ]g, and fIt; t 2 [0; T ]g de ned by (2.4), (2.5), and (2.6) is a weak solution of the normalized lter equation. De nition 2.7. The normalized lter equation has the property of pathwise uniqueness when the following holds: If f(~ ; ~ F ; f ~ Ftg; ~ P ); (~ 1 t ; ~ It)g and f(~ ; ~ F ; f ~ Ftg; ~ P ); (~ 2 t ; ~ It)g are weak solutions of the normalized lter equation such that ~ P (~ 1 0 = ~ 2 0) = 1, then ~ P ~ 1 t = ~ 2 t 8t 2 [0; T ] = 1: Remark 2.8. For the next de nition we shall need the following notation: if is a measurable mapping from some probability space (~ ; ~ F; ~ P ) into a separable metric space E, then L ~ P ( ) is the probability measure on the Borel -algebra B(E) de ned by L ~ P ( )( ) := ~ P f 2 g for each 2 B(E). De nition 2.9. The normalized lter equation has the property of uniqueness in joint law when the following holds: If f(~ ; ~ F; f ~ Ftg; ~ P ); (~ t; ~ It)g and f( ; F ; f Ftg; P ); ( t; It)g are weak solutions of the normalized lter equation such that L ~ P (~ 0) = L P ( 0), then the processes f(~ t; ~ It); t 2 [0; T ]g and f( t; It); t 2 [0; T ]g have the same nite-dimensional distributions. Remark 2.10. Under certain conditions one can associate a simpler unnormalized lter equation with the normalized lter equation. For this purpose the following additional notation is useful: If (~ ; ~ F; f ~ Ftg; ~ P ) is a complete ltered probability space, f ~ Mtg is a continuous f ~ Ftgsemimartingale, and f~ tg is a locally bounded f ~ Ftg-progressively measurable process, then ~ ~ M denotes the stochastic integral of ~ with respect to ~ M . Also, put E( ~ M )t := exp ~ Mt 1 2 h ~ Mit : 5 Now let f(~ ; ~ F; f ~ Ftg; ~ P ); (~ t; ~ It)g be a weak solution of the normalized lter equation, and de ne ~ Y k t := ~ Ik t + Z t 0 ~ shk ds; 8 t 2 [0; T ]; k = 1; : : : ; d1; (2.13) ~ t := E d1 Xk=1(~ hk) ~ Ik!t ; 8 t 2 [0; T ]: (2.14) Since f~ It; t 2 [0; T ]g is a f ~ Ftg-Wiener process, it follows that f~ t; t 2 [0; T ]g is a continuous strictly-positive f ~ Ftg-local martingale on (~ ; ~ F ; ~ P ), and 1 ~ t = E d1 Xk=1(~ hk) ~ Y k!t ; 8 t 2 [0; T ]: (2.15) De ne the M+(Rd)-valued process f ~ t; t 2 [0; T ]g on (~ ; ~ F; ~ P ) by ~ t := ~ t ~ t ; 8 t 2 [0; T ]; 2 B(Rd): (2.16) Hence ~ t := (~ t ) E d1 Xk=1(~ hk) ~ Y k!t ; 8 t 2 [0; T ]; 2 B(Rd); (2.17) and, in light of (2.11), we see that ~ P Z T 0 d1 Xk=1[~ sjhk +Bk j]2 ds <1! = 1; 8 2 C1 c (Rd) [ f1g: Using Itô's formula and the relation (2.12), we easily arrive at the Duncan-Mortensen-Zakai equation or unnormalized lter equation: for each 2 C1 c (Rd) [ f1g we have ~ t = ~ 0 + Z t 0 ~ s(A ) ds + d1 Xk=1 Z t 0 ~ s(hk +Bk ) d~ Y k s ; 8 t 2 [0; T ]: (2.18) Remark 2.11. From Remark 2.3 and (2.17) we see that t! ~ t : [0; T ]! R is continuous for each bounded continuous : Rd! R, thus f~ tg is a continuousM+(Rd)-valued process which is fFtg-adapted. Moreover, from (2.17), we see that the random element ~ 0 takes values in P(Rd), the set of probability measures on Rd. Remark 2.12. If, in (2.17), we use the optimal lter f tg in place of f~ tg and the observation process fYtg in place of f~ Ytg to get an M+(Rd)-valued and fFY t g-adapted process f tg, namely t := ( t ) E d1 Xk=1( hk) Y k!t ; 8 t 2 [0; T ]; 2 B(Rd); (2.19) 6 then f tg is called the unnormalized optimal lter for the ltering problem given by (1.1) and (1.2). Remark 2.13. In (2.18) the \driving process" f~ Ytg is the continuous f ~ Ftg-semimartingale dened on (~ ; ~ F ; ~ P ) by (2.13). The equation (2.18) becomes more tractable if we can replace ~ P with some equivalent probability measure ~ Q such that f~ Yt; t 2 [0; T ]g is an f ~ Ftg-Wiener process with respect to ~ Q. To this end, observe from (2.14) that f(~ t; ~ Ft); t 2 [0; T ]g is a continuous local martingale on (~ ; ~ F ; ~ P ), and that, if it is a martingale, then ~ Q(A) := E ~ P [ ~ T ;A]; 8A 2 ~ F ; (2.20) de nes a probability measure ~ Q on (~ ; ~ F) which is equivalent to the probability measure ~ P , namely ~ P ~ Q [ ~ F ]: (2.21) From (2.13), (2.14), and the Girsanov theorem, it then follows that f( ~ Yt; ~ Ft); t 2 [0; T ]g is a Wiener process on (~ ; ~ F ; ~ Q). Remark 2.14. A su cient condition on the weak solution f(~ ; ~ F ; f ~ Ftg; ~ P ); (~ t; ~ It)g and sensor function h( ) which ensures f(~ t; ~ Ft); t 2 [0; T ]g is a martingale on (~ ; ~ F ; ~ P ) is that E ~ P "exp 1 2 d1 Xk=1 Z T 0 [~ shk]2 ds!# <1 (see Corollary 3.5.13 of Karatzas and Shreve [6]). In particular, this condition always holds when hk 2 B(Rd), k = 1; : : : ; d1. With the preceding discussion in mind, we next formulate the notion of weak solution of the unnormalized lter equation, pathwise uniqueness and uniqueness in law (compare with D e nition IV.1 of Szpirglas [13]): De nition 2.15. A pair f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ t; ~ Yt)g is a weak solution of the unnormalized lter equation when 1. (~ ; ~ F ; f ~ Ftg; ~ Q) is a complete ltered probability space; 2. f~ Yt; t 2 [0; T ]g is an Rd1-valued f ~ Ftg-Wiener process; 3. f~ t; t 2 [0; T ]g is a M+(Rd)-valued continuous f ~ Ftg-adapted process such that the random element ~ 0 takes values in P(Rd), and, for each 2 C1 c (Rd) [ f1g, we have the following: (a) ~ Q Z T 0 d1 Xk=1[~ sjhk +Bk j]2 ds <1! = 1; (2.22) 7 (b) the LHS and RHS of (2.18) are indistinguishable. De nition 2.16. The unnormalized lter equation has the property of pathwise uniqueness when the following holds: If f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ 1 t ; ~ Yt)g and f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ 2 t ; ~ Yt)g are weak solutions of the unnormalized lter equation such that ~ Q(~ 1 0 = ~ 2 0) = 1, then ~ Q ~ 1 t = ~ 2 t 8t 2 [0; T ] = 1: De nition 2.17. The unnormalized lter equation has the property of uniqueness in joint law when the following holds: If f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ t; ~ Yt)g and f( ; F ; f Ftg; Q); ( t; Yt)g are weak solutions of the unnormalized lter equation such that L ~ Q(~ 0) = L Q( 0), then f(~ t; ~ Yt); t 2 [0; T ]g and f( t; Yt); t 2 [0; T ]g have the same nite-dimensional distributions. In this paper our goal is to establish pathwise uniqueness for the unnormalized lter equation and uniqueness in joint law for both the normalized and unnormalized lter equations. To this end we postulate the following conditions on the mappings b( ), B( ), c( ) in (1.1), and the mapping h( ) in (1.2): Condition 2.18. The mappings b : Rd ! Rd, B : Rd ! Rd d1, and c : Rd ! Rd d2 are continuous and there exists a constant C 2 [0;1) such that max i;j;k fjbi(x)j; jBij(x)j; jcik(x)jg C[1 + jxj]; 8x 2 Rd: Condition 2.19. The mapping c : Rd ! Rd d2 is such that the matrix c(x)cT (x) is strictly positive de nite for every x 2 Rd. Condition 2.20. The mapping h : Rd! Rd1 is continuous and uniformly bounded. We can now state our main result: Theorem 2.21. Suppose that Conditions 2.18, 2.19, and 2.20 hold for the nonlinear ltering problem given by E.1, E.2 and E.3. Then: (i) The unnormalized lter equation has the property of pathwise uniqueness; (ii) The unnormalized lter equation has the property of uniqueness in joint law; (iii) The normalized lter equation has the property of uniqueness in joint law. Remark 2.22. Szpirglas [13] establishes pathwise uniqueness and uniqueness in law for the normalized and unnormalized lter equations corresponding to the following nonlinear ltering problem: The signal fXtg is a homogeneous Markov process with values in a complete separable metric space E and weak in nitesimal generator A, the observation process is Yt := Wt + Z t 0 h(Xu) du; t 2 [0; T ]; where fWtg is an Rd1-valued Wiener process independent of the Markov process fXtg, and the sensor function h : E ! Rd1 is uniformly bounded and B(E)-measurable. In this context, by a 8 weak solution of the unnormalized lter equation is meant a pair f(~ ; ~ F; f ~ Ftg; ~ Q); (~ t; ~ Yt)g such that (a) (~ ; ~ F; f ~ Ftg; ~ Q) is a complete ltered probability space; (b) f~ Yt; t 2 [0; T ]g is an Rd1-valued f ~ Ftg-Wiener process; (c) f~ t; t 2 [0; T ]g is a M+(E)-valued, corlol f ~ Ftg-adapted process, the random element ~ 0 takes values in P(E), and supt2[0;T ]E[j~ t1j2] <1; (d) For each 2 D(A) [the domain of the generator A] one has to within indistinguishability that ~ t = ~ 0 + Z t 0 ~ s(A ) ds + d1 Xk=1 Z t 0 ~ s(hk ) d~ Y k s ; 8 t 2 [0; T ]: (2.23) (See D e nition IV.1 of Szpirglas [13]). The nice thing about (2.23) is that it includes reference to just one unbounded linear operator, namely the in nitesimal generator A of the signal process, and the resolvent identity can be used to eliminate A and re-write (2.23) in the form ~ t = ~ 0(Pt ) + d1 Xk=1 Z t 0 ~ s(hkPt s )d~ Y k s ; 8 t 2 [0; T ]; (2.24) where fPtg is the Borel semigroup with in nitesimal generator A. There is complete equivalence between (2.23) and (2.24) in the sense that if the pair f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ t; ~ Yt)g is subject to (a), (b), (c), then (2.23) holds for each 2 D(A) if and only if (2.24) holds for each 2 B(E) (see Th eor eme IV.1 of [13]). Consequently, it is enough to establish pathwise uniqueness for (2.24) in order to conclude pathwise uniqueness for the unnormalized lter equation. The advantage of (2.24) is that it involves only the bounded linear operators fPtg, and this structure makes it possible to establish pathwise uniqueness for solutions of (2.24) by iterating a simple integral inequality (see Section V.2 of Szpirglas [13]). Comparing (2.23) with the unnormalized lter equation (2.18) for the nonlinear ltering problem de ned by (1.1) and (1.2), we see that (2.18) includes two unbounded linear operators, namely the rst-order di erential operator Bk which results from dependence of the signal fXtg on the Wiener process fWtg of the observation equation, as well as the second-order di erential operator A corresponding to the signal process fXtg. In this case there seems to be no clear way of adapting the elegant semigroup ideas of [13] to remove both of these unbounded operators and get an equivalent equation involving just bounded linear operators. Accordingly, the approach that we shall use to establish Theorem 2.21(i) is di erent from that of Szpirglas [13], and relies on a uniqueness theorem for measurevalued evolution equations (see Theorem 3.30 to follow). Remark 2.23. Uniqueness for the normalized and unnormalized lter equations has also been studied by Bhatt, Kallianpur and Karandikar [1], Kurtz and Ocone [8], and Rozovskii [11] from a somewhat di erent point of view than that taken by Szpirglas [13] and the present work. To see this in the context of the ltering problem given by (1.1) and (1.2), observe from Remark 2.12 that the unnormalized optimal lter f tg solves the Duncan-Mortensen-Zakai equation, namely 9 for each 2 C1 c (Rd) [ f1g we have t = 0 + Z t 0 s(A ) ds+ d1 Xk=1 Z t 0 s(hk +Bk ) dY k s ; 8 t 2 [0; T ]: (2.25) With this in mind, the following question is natural: Suppose that f tg is someM+(Rd)-valued, corlol, and fFY t g-adapted process on ( ;F ; P ), such that for each 2 Cc(Rd) [ f1g we have t = 0 + Z t 0 s(A ) ds + d1 Xk=1 Z t 0 s(hk +Bk ) dY k s ; 8 t 2 [0; T ]: (2.26) Does it follow that f tg and f tg are indistinguishable? The works of Bhatt, Kallianpur and Karandikar ([1], Theorem 3.1), Kurtz and Ocone ([8], Theorems 4.2 and 4.7), and Rozovskii ([11], Theorem 3.1) provide conditions on the nonlinear ltering problem for which the answer is in the a rmative. Uniqueness in this sense is useful for the following reason: the observation process fYtg is the random data that \drives" the unnormalized lter equation (2.25), and if we can \non-anticipatively" use the individual paths of fYtg to compute a measure-valued process f tg which satis es (2.26) by some numerical method for example then uniqueness ensures that f tg is in fact the desired unnormalized optimal lter f tg. It should be noted that uniqueness in the preceding sense can be established for much more general nonlinear ltering problems than that represented by the simple model (1.1) and (1.2). In fact, Theorem 3.1 of [1] deals with a ltering problem in which the signal process takes values in a complete separable metric space (not necessarily locally compact), the sensor function h( ) need not be uniformly bounded but only satisfy a nite-energy condition similar to (2.3), the dependence of the signal fXtg on the Wiener process fWtg is more general than that given by the explicit model (1.1), (1.2) (see (1.3) of [1]), and the joint signal/observation process f(Xt; Yt)g is the corlol solution of a well-posed martingale problem. On the other hand, the uniqueness question dealt with in [1], [8] and [11] is rather di erent from that addressed by Theorem 2.21, since the candidate solution f tg is postulated to be adapted speci cally to the observation ltration fFY t g. In contrast, Theorem 2.21 establishes pathwise uniqueness in a sense closer to that formulated for Itô stochastic di erential equations, where the candidate solutions being compared should be de ned on an arbitrary ltered probability space. Remark 2.24. A basic property of Itô SDE's due to Yamada andWatanabe [14] is that pathwise uniqueness implies uniqueness in joint law, so that pathwise uniqueness is the stronger of the two uniqueness properties. It is shown in Szpirglas [13] that the basic Yamada-Watanabe argument extends to the measure-valued lter equations, so that pathwise uniqueness is again the stronger property (this is how we will conclude (ii) and (iii) from (i) in Theorem 2.21). Linearity of the unnormalized lter equation in fact implies the converse, so that for this equation the two uniqueness properties are actually equivalent: Theorem 2.25. Suppose that Conditions 2.18, 2.19, and 2.20 hold for the nonlinear ltering problem given by E.1, E.2 and E.3. Then uniqueness in joint law implies pathwise uniqueness. 10 3 Proofs of Theorems 2.21 and 2.25: Proof of Theorem 2.21(i) We shall need the following result, the proof of which is given in Section 5: Fact 3.26. Suppose that Conditions 2.18-2.20 hold and let f(~ ; ~ F; f ~ Ftg; ~ Q); (~ ; ~ Y )g be a weak solution of the unnormalized lter equation. Then, for every 2 (1;1) there exists a constant ( ) 2 [0;1) such that E ~ Q sup 0 s T j~ s1j ( ): (3.27) Now x two weak solutions f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ i t; ~ Yt)g; i = 1; 2, of the unnormalized lter equation, such that ~ Q ~ 1 0 = ~ 2 0 = 1; (3.28) and de ne product measures on (R2d;B(R2d)) by 12 t ( ; ~ !) := (~ 1 t ~ 2 t )( ; ~ !); 8(t; ~ !) 2 [0; T ] ~ : A simple application of the Dynkin class theorem establishes Fact 3.27. For every 2 B(R2d), the mapping (t; ~ !) ! 12 t ( ; ~ !) : ~ [0; T ] 7! [0;1) is measurable with respect to the f ~ Ftg-progressive -algebra. Also put 12 t ( ) := E ~ Q[ 12 t ( )]; 8 2 B(R2d); t 2 [0; T ]: (3.29) It readily follows that 12 t de nes a positive measure on (R2d;B(R2d)) for every t 2 [0; T ]. By Fact 3.26, 12 t (R2d) = E ~ Q[(~ 1 t 1)(~ 2 t 1)] E ~ Q[ sup 0 s T j~ 1 s1j2] 1 2 E ~ Q[ sup 0 s T j~ 2 s1j2] 1 2 (2); 8t 2 [0; T ]: (3.30) This shows that 12 t is a positive measure on (R2d;B(R2d), uniformly bounded with respect to t 2 [0; T ], while Fact 3.27 with Fubini's theorem shows that the mapping t! 12 t ( ) : [0; T ] 7! R is Borel-measurable for each 2 B(R2d). Next, de ne 11 t ; 22 t 2 M+(R2d); t 2 [0; T ], analogously to 12 t , by ii t ( ) := E ~ Q[(~ i t ~ i t)( )]; 8 2 B(R2d); t 2 [0; T ]; i = 1; 2: (3.31) In the same way as for 12 , we see that ii are positive measures on (R2d;B(R2d)), uniformly bounded with respect to t 2 [0; T ], and the mappings t ! ii t ( ) : [0; T ] 7! R are Borelmeasurable for each 2 B(R2d), i = 1; 2. 11 Remark 3.28. For mappings f1; f2 2 B(Rd) de ne the tensor product of f1 with f2 to be the mapping f1 f2 : R2d! R given by f1 f2(x1; x2) := f1(x1)f2(x2); 8 x1; x2 2 Rd: In view of (3.29) and (3.31), for each f1; f2 2 B(Rd) we have 12 t (f1 f2) = E ~ Q[(~ 1 t f1)(~ 2 t f2)]; (3.32) ii t (f1 f2) = E ~ Q[(~ i tf1)(~ i tf2)]; i = 1; 2: (3.33) From (3.28), (3.29), and (3.31) we see that 11 0 = 22 0 = 12 0 : (3.34) Using this fact, we shall establish 11 t = 22 t = 12 t ; 8t 2 [0; T ]; (3.35) from which pathwise uniqueness follows. Indeed, if (3.35) holds, then for each f 2 B(Rd) we have 11 t (f f) = 22 t (f f) = 12 t (f f); 8t 2 [0; T ]; and therefore from (3.32) and (3.33), E ~ Q[(~ 1 tf ~ 2 t f)2] = E ~ Q[(~ 1 t f)(~ 1 t f)] 2E ~ Q[(~ 1 t f)(~ 2 t f)] + E ~ Q[(~ 2 tf)(~ 2 t f)] = 11 t (f f) 2 12 t (f f) + 22 t (f f) = 0: Thus, for each t 2 [0; T ] and f 2 B(Rd), we have ~ Q ~ 1 t f = ~ 2 t f = 1: (3.36) Now Ĉ(Rd) equipped with the supremum norm k k is separable. Thus, from (3.36), for each t 2 [0; T ] there is a ~ Q-null event Nt 2 ~ F such that, for each ~ ! 62 Nt, we have ~ 1 t (~ !)f = ~ 2 t (~ !)f; 8 f 2 Ĉ(Rd): (3.37) But Ĉ(Rd) separates bounded positive measures on B(Rd) (see Problem 5.4.25 of Karatzas and Shreve [6]), thus (3.37) establishes ~ Q[~ 1 t = ~ 2 t ] = 1 for each t 2 [0; T ]. Now Theorem 2.21(i) follows from the fact that f~ i t; t 2 [0; T ]g are continuous (recall De nition 2.15). It therefore remains to establish (3.35) in order to prove Theorem 2.21(i). To this end, for each x1; x2 2 Rd de ne the 2d 2d matrix a(x1; x2), the 2d vector b(x1; x2), and the real number 12 h(x1; x2) by a(x1; x2) := "ccT (x1) 0 0 ccT (x2)#+ "B(x1) B(x2)# hBT (x1) BT (x2)i (3.38a) b(x1; x2) := "b(x1) +B(x1)h(x2) b(x2) +B(x2)h(x1)# (3.38b) h(x1; x2) := d1 Xk=1 hk(x1)hk(x2): (3.38c) Observe that the matrix a(x1; x2) is symmetric and strictly positive-de nite (see Condition 2.19), and let A be the second order linear di erential operator corresponding to the matrices a and b, namely A (x) := 2d Xi=1 bi(x)@i (x) + 1 2 2d X i;j=1 aij(x)@i@j (x); 8 2 C1(R2d); x 2 R2d: (3.39) From (3.38a), (3.38b), Condition 2.18, and Condition 2.20, there is a constant K 2 [0;1) such that max i j bi(x)j K[1 + jxj]; max i;j j aij(x)j K[1 + jxj2]; 8x 2 R2d; (3.40) and the operator A has the following property, which is established in Section 4: Lemma 3.29. Suppose that Conditions 2.18{2.20 hold. Then the M+(R2d)-valued functions 11 ; 12 , and 22 given by (3.29) and (3.31) solve the evolution equation tf = 0f + Z t 0 s( Af + hf) ds; 8t 2 [0; T ]; 8f 2 spanf1; C1 c (R2d)g: (3.41) It remains to show that uniqueness holds for the measure-valued evolution equation (3.41), since this fact along with (3.34) gives (3.35), as required. To this end we shall use the following basic uniqueness theorem due to Bhatt and Karandikar (see Theorem 3.4 and Remark 1 of [3]): Theorem 3.30 (Bhatt and Karandikar). Suppose that E be a complete separable metric space, that 2 C(E), and that L : D(L)! C(E) is a linear operator with domain D(L) C(E), such that the following conditions hold: C1: D(L) is an algebra that separates points of E, and contains constants with L1 = 0; C2: there exists a countable subset ffng D(L) such that bp-closuref(fn;Lfn) : n 2 Ng f(f;Lf) : f 2 D(L)g; 13 C3: every progressively measurable solution of the martingale problem for L has a corlol modication; C4: for every x 2 E the (corlol) martingale problem for (L; x) is well-posed. Under these conditions we have the following: If i : [0; T ] 7! M+(E); i = 1; 2 are such that 1. 10 = 20; 2. for every 2 B(E), the mappings t 7! it( ); i = 1; 2, are Borel-measurable on [0; T ]; 3. itf = i0f + R t 0 is(Lf + f) ds; 8t 2 [0; T ]; 8f 2 D(L); i = 1; 2; then 1t = 2t ; 8t 2 [0; T ]: Remark 3.31. Since Lf 2 C(E), 8 f 2 D(L), we can take 1 in (2.1) and the separability condition (I) on pages 328 and 329 of [3]. Our statement of Theorem 3.30 includes these simpli cations. We are going to verify the conditions of Theorem 3.30 for E := R2d, := h, and D(L) := spanf1; C1 c (R2d)g; Lf := Af; 8f 2 D(L): ) (3.42) From (3.42) we see that D(L) is an algebra that separates points and contains constants, and (3.39) ensures L1 = 0, so that C1 is veri ed. To check C2, de ne L := spanf1; Ĉ(R2d)g, and observe that L, equipped with the supremum norm, is homeomorphic to the separable space C(R2d ) [where R2d denotes the one-point compacti cation of R2d], and is therefore separable. Therefore, the graph of L is separable (as a subset of the separable space L L), and C2 follows (c.f. Remark 2.5 of Kurtz [7]). To check C3, observe from (3.42) that a solution of the martingale problem for L is a solution of the martingale problem for ( A; C1 c (R2d)), and Proposition 5.3.5 of Ethier and Kurtz ensures that a solution of the martingale problem for ( A; C1 c (R2d) has a continuous, hence corlol, modi cation. Finally, to check C4, observe from Theorem 8.1.7 of Ethier and Kurtz [4], along with (3.40) and strict positive-de niteness of a(x), 8 x 2 R2d, (which is secured by Condition 2.19), that the martingale problem for the operator ( A; C1 c (R2d)) is well-posed, and therefore the martingale problem for L is well-posed. Now (3.35) follows from Theorem 3.30, (3.34), the established Borel-measurability of the mappings t ! 11 t ( ), t ! 12 t ( ) and t ! 22 t ( ) for each 2 B(R2d), and Lemma 3.29. This establishes Theorem 2.21(i). Remark 3.32. The proof just given relies on the special structure of the unnormalized lter equation (2.18) and does not appear to extend to the normalized lter equation (2.10). We have therefore not been able to establish pathwise uniqueness in the sense of De nition 2.7 under conditions comparable to those of Theorem 2.21. 14 Proof of Theorem 2.21(iii): Let f(~ ; ~ F; f ~ Ftg; ~ P ); (~ t; ~ It)g and f( ; F ; f Ftg; P ); ( t; It)g be two weak solutions of the normalized lter equation. By an argument similar to that used for Proposition IX.1.4 of Revuz and Yor [9], to establish uniqueness in joint law it is enough to show that the processes f(~ t; ~ It)g and f( t; It)g are identically distributed when ~ 0 = 0 = ; for each 2 P(Rd). (3.43) Thus suppose (3.43) holds for some 2 P(Rd). Put ~ t := E( d1 Xk=1(~ hk) ~ Ik)t and t := E( d1 Xk=1( hk) Ik)t; 8t 2 [0; T ]; and de ne the measures ~ Q and Q on the measurable spaces (~ ; ~ F) and ( ; F) respectively by ~ Q(A) := E ~ P [ ~ T ;A]; 8A 2 ~ F ; (3.44) Q(A) := E P [ T ;A]; 8A 2 F: (3.45) Then, with ~ Yt := ~ It + d1 Xk=1 Z t 0 ~ uhk du; Yt := It + d1 Xk=1 Z t 0 uhk du; t 2 [0; T ]; (3.46) and ~ t := ~ t=~ t; t := t= t; 8t 2 [0; T ]; (3.47) we see, as in Remark 2.10 and Remark 2.13, that the pairs f(~ ; ~ F; f ~ Ftg; ~ Q); (~ t; ~ Yt)g and f( ; F ; f Ftg; Q); ( t; Yt)g are weak solutions of the unnormalized lter equation, with ~ 0 = 0 = : For a complete separable metric space E, let CE[0; T ] denote the complete separable metric space of all continuous mappings from [0; T ] into E with the usual metric giving uniform convergence over [0; T ]. De ne ̂ := CM+(Rd)[0; T ] CM+(Rd)[0; T ] CRd1[0; T ]; which is a complete separable metric space with the usual product metric, and let !̂ = (!1; !2; !3) be a generic member of ̂. By the Yamada-Watanabe construction (see Theorem IV.1.1 of Ikeda and Watanabe [5]), there exists P̂ 2 P( ̂) such that YW.1: LP̂ (!1; !3) = L ~ Q(~ ; ~ Y ); YW.2: LP̂ (!2; !3) = L Q( ; Y ); 15 YW.3: If ( ̂; F̂; P̂ ) is the completion of ( ̂;B( ̂); P̂ ), and F̂t is the augmentation of the -algebra f!̂(s); s 2 [0; t]g with the null events of ( ̂; F̂ ; P̂ ), 8 t 2 [0; T ], then f!3 t ; t 2 [0; T ]g is a fF̂tg-Wiener process on ( ̂; F̂; P̂ ). From (YW.1), (YW.2), and (YW.3), along with Exercise IV.5.16 of Revuz and Yor [9], it follows that f( ̂; F̂ ; fF̂tg; P̂ ); (!1; !3)g and f( ̂; F̂ ; fF̂tg; P̂ ); (!2; !3)g are weak solutions for the unnormalized lter equation with !1 0 = !2 0 = ; and hence, from Theorem 2.2(i), P̂ (!1 t = !2 t 8t 2 [0; T ]) = 1: (3.48) From (3.47) we see that ~ t = (~ t )(~ t1); 8 t 2 [0; T ]; 2 B(Rd); (3.49) and so, from (3.46), ~ Ik t = ~ Y k t d1 Xk=1 Z t 0 (~ u1)(~ uhk) du; 8 k = 1; 2; : : : ; d1; t 2 [0; T ]: (3.50) From (3.49) and (3.50) there exists a measurable mapping : CM+(Rd)[0; T ] CRd1[0; T ] ! CP(Rd)[0; T ] CRd1[0; T ] such that (~ ; ~ I) = (~ ; ~ Y ): (3.51) Now (3.49) and (3.50) continue to hold with \overbar" in place of \tilde", and hence ( ; I) = ( ; Y ): (3.52) Thus, for each 2 B(CP(Rd)[0; T ] CRd1[0; T ]), we see from (3.44), (3.51), and (YW.1), that ~ P ((~ ; ~ I) 2 ) = E ~ Q[(~ T1) 1I ( (~ ; ~ Y ))] = EP̂ [(!1 T1) 1I ( (!1; !3))]; (3.53) and, from (3.45), (3.52), and (YW.2), we similarly have P (( ; I) 2 ) = E Q[( T1) 1I ( ( ; Y ))] = EP̂ [(!2 T1) 1I ( (!2; !3))]: (3.54) Now (3.48), (3.53), and (3.54) show that ~ P ((~ ; ~ I) 2 ) = P (( ; I) 2 ), as required. Proof of Theorem 2.21(ii): The proof is an obvious simpli cation of the proof of Theorem 2.21(iii) and is omitted. 16 Proof of Theorem 2.25: Let f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ i t; ~ Yt)g; i = 1; 2 be two weak solutions of the unnormalized lter equation. De ne ~ 3 t ( ) := ~ 1 t ( ) + ~ 2 t ( ) 2 ; t 2 [0; T ]: It then follows that f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ 3 t ; ~ Yt)g is a weak solution of the unnormalized lter equation. Therefore, the postulated uniqueness in joint law together with Fact 3.26 implies that for an arbitrary 2 C1 c (Rd) [ f1g we have 2E ~ Q " ~ 1 t + ~ 2 t 2 2# E ~ Q[(~ 1 t )2] E ~ Q[(~ 2 t )2] = 0; 8t 2 [0; T ]: Rearanging this expression gives E ~ Q[(~ 1 t ~ 2 t )2] = 0; 8t 2 [0; T ]: (3.55) Since Ĉ(Rd) is separable (in the supremum norm), it follows that C1 c (Rd) is likewise separable, and therefore, by Problem 5.4.25 of Karatzas and Shreve [6], there is a countable determining set for M+(Rd) in C1 c (Rd) [ f1g. Then (3.55) shows that f~ 1g and f~ 2g are modi cations of each other, hence indistinguishable (since f~ 1g and f~ 2g are continuous). 4 Proof of Lemma 3.29 For arbitrary f1; f2 2 C1 c (Rd) put ~ A(f1 f2) := f1 (Af2) + (Af1) f2 + d1 Xk=1 (hkf1) (hkf2) + (hkf1) (Bkf2) + (Bkf1) (hkf2) + (Bkf1) (Bkf2) ; (4.56) where A and Bk are given by (2.7). We now have the following two lemmas, the proofs of which are deferred to Section 5: Lemma 4.33. Suppose Conditions 2.18{2.20 hold, let f(~ ; ~ F ; f ~ Ftg; ~ Q); (~ i t; ~ Yt)g, i = 1; 2; be weak solutions of the unnormalized lter equation, and de ne the M+(R2d)-valued functions 12 ; 11 and 22 as in (3.29) and (3.31). Then, for each f1; f2 2 C1 c (Rd), we have 12 t (f1 f2) = 12 0 (f1 f2) + Z t 0 12 u ( ~ A(f1 f2)) du; t 2 [0; T ]; (4.57) with identical relations for 11 and 22 in place of 12 . Lemma 4.34. For ~ A and A de ned in (4.56) and (3.39) respectively, we have ~ A(f1 f2)(x) = A(f1 f2)(x) + h(x)(f1 f2)(x); 8 x 2 R2d; (4.58) for each f1; f2 2 C1 c (Rd). 17 De ne ~ D := spanff1 f2 : f1; f2 2 C1 c (Rd)g: (4.59) Putting Lemma 4.34 and Lemma 4.33 together, we see that the M+(R2d)-valued functions 12 , 11 , and 22 , de ned at (3.29) and (3.31), each solve the evolution equation tf = 0f + Z t 0 u( Af + hf) du; t 2 [0; T ]; 8f 2 ~ D: (4.60) In order to prove Lemma 3.29, it remains to show that this relation holds not only for f 2 ~ D, but for all f in the larger domain spanf1; C1 c (R2d)g. To this end we need the following result, which is established later in this section: Lemma 4.35. Suppose Conditions 2.18{2.20 hold. Then the closure of the relation f(f; Af); f 2 ~ Dg in the supremum norm of B(R2d) B(R2d) contains the relation f(f; Af); f 2 C1 c (R2d)g. From Lemma 4.35 and the notions of bp-closedness and bp-closure of a relation (see foot of page 111 of Ethier and Kurtz [4]), we see that f(f; Af); f 2 C1 c (R2d)g bp-closuref(f; Af); f 2 ~ Dg: (4.61) Now put S12 := (f; g) 2 B(R2d) B(R2d) : 12 t f = 12 0 f + Z t 0 12 s (g + hf) ds; 8t 2 [0; T ] ; (4.62) and observe that S12 is a linear relation. By (3.30) we have sup 0 t T 12 t (R2d) <1; and therefore, since h 2 B(R2d), it follows from the dominated convergence theorem that the linear relation S12 is bp-closed in B(R2d) B(R2d). Since the M+(R2d)-valued function 12 solves the evolution equation (4.60), we have f(f; Af); f 2 ~ Dg S12, and therefore, from the bp-closedness of S12 and (4.61), we have f(f; Af); f 2 C1 c (R2d)g S12: (4.63) Next, observe from (3.40) and Problem 4.11.12 of Ethier and Kurtz [4] that the operator ( A; C1 c (R2d)) is conservative, and hence (see page 166 of [4]) we have (1; 0) 2 bp-closuref(f; Af); f 2 C1 c (R2d)g: (4.64) In the light of (4.64), (4.63), and the bp-closedness of S12, we then get (1; 0) 2 S12: (4.65) 18 Now (4.63), (4.65), and linearity of the relation S12 shows that f(f; Af); f 2 spanf1; C1 c (R2d)gg S12; which, in view of (4.62), shows that 12 solves the evolution equation (3.41). De ning S11 as in (4.62), but with 11 in place of 12 , we can similarly show that 11 solves (3.41), and likewise for 22 . 5 Proofs of Technical Results Proof of Fact 3.26: Fix some 2 (1;1). Since ~ 0 takes values in P(Rd), we see from (2.18) with 1 that ~ t1 = 1 + Z t 0 d1 Xk=1 ~ shk ~ s1 ( s1) d~ Y k s : This gives (see Exercise IV.3.10(1) of Revuz and Yor [9]) ~ t1 = E d1 Xk=1 ~ hk ~ 1 ~ Y k !t ; (5.66) and hence j~ t1j = ~ Mt exp ( 1) 2 d1 Xk=1 Z t 0 ~ shk ~ s1 2 ds! ~ Mt exp ( 1) 2 khkT ; 8 t 2 [0; T ]; (5.67) for ~ Mt := E d1 Xk=1 ~ hk ~ 1 ~ Y k !t and khk := sup x2Rd jh(x)j: Now Condition 2.20 ensures that the processes f(~ thk)=(~ t1); t 2 [0; T ]g are uniformly bounded (by khkk), and therefore f( ~ Mt; ~ Ft); t 2 [0; T ]g is a continuous martingale on (~ ; ~ F ; ~ Q), with ~ M0 = 1. Taking ~ Q-expectations in (5.67) then gives E ~ Q[j~ t1j ] exp ( 1) 2 khkT ; 8 t 2 [0; T ]: (5.68) Again, by (5.66) and uniform-boundedness of the processes f(~ thk)=(~ t1); t 2 [0; T ]g we see that f(~ t1; ~ Ft); t 2 [0; T ]g is a continuous martingale on (~ ; ~ F ; ~ Q), which, in light of (5.68), 19 is L -bounded. Thus, by Doob's inequality, there is some ( ) 2 (0;1) such that (3.27) holds. Proof of Lemma 4.33: Fix f1; f2 2 C1 c (Rd); t 2 [0; T ]. Since f~ ; ~ F ; f ~ Ftg; ~ Q); (~ i t; ~ Yt)g; i = 1; 2 are weak solutions of the unnormalized lter equations, we have ~ i tfi = ~ i 0fi + Z t 0 ~ i u(Afi) du+Xk Z t 0 ~ i u(hkfi +Bkfi) d~ Y k u ; i = 1; 2: (5.69) By Itô's formula from (5.69) we have (~ 1 t f1)(~ 2 t f2) = (~ 1 0f1)(~ 2 0f2) + Z t 0 (~ 1 uf1) d(~ 2 f2)u + Z t 0 (~ 2 uf2) d(~ 1 f1)u + h~ 1f1; ~ 2f2it; (5.70) hence (~ 1 t f1)(~ 2 t f2) = (~ 1 0f1)(~ 2 0f2) + Z t 0 (~ 1 uf1)(~ 2 u(Af2)) du+ Z t 0 (~ 2 uf2)(~ 1 u(Af1)) du +Xk Z t 0 (~ 1 u(hkf1))(~ 2 u(hkf2)) + (~ 1 u(hkf1))(~ 2 u(Bkf2)) + (~ 1 u(Bkf1))(~ 2 u(hkf2)) + (~ 1 u(Bkf1))(~ 2 u(Bkf2)) du +Mt; (5.71) whereMt :=Xk Z t 0 (~ 1 uf1)[~ 2 u(hkf2 +Bkf2)] + (~ 2 uf2)[~ 1 u(hkf1 +Bkf1)] d~ Y k u ; t 2 [0; T ]; is a f ~ Ftg-continuous local martingale on (~ ; ~ F; ~ Q). Observe by Fact 3.26 and Cauchy-Schwarz thatE ~ Q Z t 0 j~ 1 uf1j2j~ 2 u(hkf2 +Bkf2)j2 du kf1k2[khkf2k+ kBkf2k]2 E ~ Q Z t 0 j~ 1 u1j2j~ 2 u1j2 du (4)Tkf1k2[khkf2k+ kBkf2k]2 <1; and so fMtg is a square-integrable f ~ Ftg-martingale on (~ ; ~ F; ~ Q), with M0 = 0. Therefore, by taking ~ Q-expectations in (5.71), we get E ~ Q[(~ 1 t f1)(~ 2 t f2)] = E ~ Q[(~ 1 0f1)(~ 2 0f2)] + Z t 0 E ~ Q[(~ 1 uf1)(~ 2 u(Af2))] du+ Z t 0 E ~ Q[(~ 1 u(Af1))(~ 2 uf2)] du +Xk Z t 0 hE ~ Q[(~ 1 u(hkf1))(~ 2 u(hkf2))] + E ~ Q[(~ 1 u(hkf1))(~ 2 u(Bkf2))] + E ~ Q[(~ 1 u(Bkf1))(~ 2 u(hkf2))] + E ~ Q[(~ 1 u(Bkf1))(~ 2 u(Bkf2))]i du; 20 and thus, from (3.32), 12 t (f1 f2) = 12 0 (f1 f2) + Z t 0 [ 12 u (f1 Af2) + 12 u (Af1 f2)] du +Xk Z t 0 [ 12 u (hkf1 hkf2) + 12 u (hkf1 Bkf2) + 12 u (Bkf1 hkf2) + 12 u (Bkf1 Bkf2)] du; which establishes (4.57). The proofs of the corresponding facts for 11 and 22 follow in exactly the same way. Proof of Lemma 4.34: Fix some f1; f2 2 C1 c (Rd), and let f := f1 f2 (recall Remark 3.28). Write x 2 R2d in the form x = (x1; x2), for x1; x2 2 Rd. We then have the rst derivatives @if(x) = f2(x2)@if1(x1); @i+df(x) = f1(x1)@if2(x2); (5.72) for i = 1; : : : ; d, and the second derivatives @i@jf(x) = f2(x2)@i@jf1(x1); @i+d@j+df(x) = f1(x1)@i@jf2(x2); @i@j+df(x) = @if1(x1)@jf2(x2); 9>=>; : (5.73) for i; j = 1; : : : ; d. From (4.56) we can put ~ A(f1 f2)(x) = I1(x) + I2(x) + h(x)f(x); (5.74) where I1(x) is the sum of all terms of ~ A(f1 f2)(x) which involve second-order derivatives of the function f (see (2.7a) and (2.7b)) namely I1(x) := 1 2 d X i;j=1[BBT + ccT ]ij(x2)f1(x1)@i@jf2(x2) + 12 d X i;j=1[BBT + ccT ]ij(x1)f2(x2)@i@jf1(x1) + d1 Xk=1 d Xi=1 Bik(x1)@if1(x1)! d Xj=1 Bjk(x2)@jf2(x2)! ; (5.75) 21 and I2(x) is the sum of all terms of ~ A(f1 f2)(x) which involve rst-order derivatives of f namely I2(x) := d Xi=1 bi(x1)f2(x2)@if1(x1) + d Xi=1 bi(x2)f1(x1)@if2(x2) + d1 Xk=1 hk(x1)f1(x1) d Xi=1 Bik(x2)@if2(x2) + d1 Xk=1 hk(x2)f2(x2) d Xi=1 Bik(x1)@if1(x1): (5.76) For each x 2 R2d put B(x) := "B(x1) B(x2)# hBT (x1) BT (x2)i ; C(x) := "ccT (x1) 0 0 ccT (x2)# ; (5.77) and observe that [ B]ij(x) = [BBT ]ij(x1); [ B](i+d)(j+d)(x) = [BBT ]ij(x2); (5.78a) [ B]i(j+d)(x) = [ B](j+d)i(x) = d1 Xk=1 Bik(x1)Bjk(x2); (5.78b) and [ C]i;j(x) = [ccT ]ij(x1); [ C](i+d)(j+d)(x) = [ccT ]ij(x2); (5.79a) [ C]i(j+d) = [ C](j+d)i = 0; (5.79b) for all i; j = 1; : : : ; d. From (5.73), (5.75), (5.78) and (5.79), I1(x) =1 2 d X i;j=1[ B + C](i+d)(j+d)(x)@i+d@j+df(x) + 1 2 d X i;j=1[ B + C]ij(x)@i@jf(x) +1 2 d X i;j=1[ B + C]i(j+d)(x)@i@j+df(x) + 1 2 d X i;j=1[ B + C](j+d)i(x)@j+d@if(x) =1 2 2d X i;j=1[ B + C]ij(x)@i@jf(x); and, from (3.38a) and (5.77), a(x) = B(x) + C(x); hence I1(x) = 1 2 2d X i;j=1 aij(x)@i@jf(x): (5.80) 22 Similarly, from (5.72), (5.76), and (3.38b), I2(x) = d Xi=1 "bi(x1) + d1 Xk=1 Bik(x1)hk(x2)# @if(x) + d Xi=1 "bi(x2) + d1 Xk=1 Bik(x2)hk(x1)# @i+df(x) = d Xi=1 bi(x)@if(x) + d Xi=1 bi+d(x)@i+df(x) = 2d Xi=1 bi(x)@if(x); (5.81) Now (4.58) follows from (3.39), (5.74), (5.80) and (5.81). Proof of Lemma 4.35: Fix arbitrary 2 (0;1) and g 2 C1 c (R2d). Put BR := fx 2 R2d : jxj Rg; R 2 [0;1); and x R such that supp(g) BR. Also x some q 2 C1 c (Rd) such that kqk 1; (5.82a) q(z) = 1; 8 z 2 Rd; with jzj R; (5.82b) q(z) = 0; 8 z 2 Rd; with jzj Rp2: (5.82c) By Proposition 7.1 in Appendix 7 of Ethier and Kurtz [4], there exists a polynomial p : R2d! R such that max x2B2R jg(x) p(x)j < ; (5.83a) max x2B2R j@ig(x) @ip(x)j < ; 8 i = 1; : : : ; 2d; (5.83b) max x2B2R j@i@jg(x) @i@jp(x)j < ; 8 i; j = 1; : : : ; 2d: (5.83c) Since g(x) = 0 when x 62 BR, we note from (5.83a) that sup x2B2RnBR jp(x)j < : (5.84) For all x 2 R2d, put x := (x1; x2), x1; x2 2 Rd, and de ne q(x) := q(x1)q(x2); f(x) := q(x) p(x): Since q 2 C1 c (Rd) and p(x) is a polynomial in x = (x1; x2), it follows that f 2 ~ D (recall (4.59)). 23 From (5.82), we have q(x) = 0 when x 62 B2R, and q(x) = 1 when x 2 BR. Thus kf gk = sup x2R2d j q(x)p(x) g(x)j = max x2B2R j q(x)p(x) g(x)j max x2BR jp(x) g(x)j+ sup x2B2RnBR j q(x)p(x) g(x)j; hence (5.83a) and (5.84) give kf gk + sup x2B2RnBR jp(x)j 2 : (5.85) Next, consider k Af Agk. From (3.39) we have Af(x) = q(x) Ap(x) + p(x) A q(x) + (rp(x))T a(x)r q(x): (5.86) By the choice of R we have g(x) = 0 and therefore Ag(x) = 0; 8x 62 BR. Moreover, from (5.82c), we have q(x) = 1, and therefore r q(x) = 0 and A q(x) = 0, 8 x 2 BR. Similarly, q(x) = 0, and therefore A q(x) = 0, 8x 62 B2R. Then, it follows from (5.86) that k Af Agk = max x2B2R j q(x) Ap(x) + p(x) A q(x) + (rp(x))T a(x)r q(x) Ag(x)j max x2BR j q(x) Ap(x) + p(x) A q(x) + (rp(x))T a(x)r q(x) Ag(x)j + sup x2B2RnBR j q(x) Ap(x) + p(x) A q(x) + (rp(x))T a(x)r q(x)j = max x2BR j Ap(x) Ag(x)j + sup x2B2RnBR(j q(x) Ap(x)j + jp(x) A q(x)j+ j(rp(x))T a(x)r q(x)j): (5.87) Since a and b are locally bounded, we have C1 := max x2B2R 2d Xi=1 j bi(x)j+ 1 2 2d X i;j=1 j aij(x)j! <1: Also let C2 := k qk+Xi k@i qk+Xi;j k@i@j qk <1: Then by (5.83) max x2BR j Ap(x) Ag(x)j < C1 : (5.88) Similarly, by (5.83) and the fact that g(x) = 0; 8x 2 B2R nBR, we obtain j@ip(x)j < ; j Ap(x)j < C1 ; ) 8 x 2 B2R nBR; 24 and hence, from these bounds and (5.84),supx2B2RnBR(jq(x) Ap(x)j+ jp(x) Aq(x)j+ j(rp(x))T a(x)rq(x)j) C1 + C1C2 + C1C2: (5.89)Now, upon combining (5.87), (5.88), and (5.89) we havekAg Afk 2 (C1 + C1C2);(5.90)and the result follows.25 References[1] A. G. Bhatt, G. Kallianpur and R. L. Karandikar. 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