On Émery s inequality and a variation-of-constants formula

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1 On Émery s inequality and a variation-of-constants formula Markus Reiß Institute of Applied Mathematics, University of Heidelberg Im Neuenheimer Feld 294, 6912 Heidelberg, Germany Markus Riedle Department of Mathematics, Humboldt University of Berlin Unter den Linden 6, 199 Berlin, Germany Onno van Gaans Mathematical Institute, Leiden University P.O. Box 9512, 23 RA Leiden, The Netherlands vangaans@math.leidenuniv.nl Abstract A generalization of Émery s inequality for stochastic integrals is shown for convolution integrals of the form ( g(t sy (s dz(s t, where Z is a semimartingale, Y an adapted càdlàg process, and g a deterministic function. An even more general inequality for processes with two parameters is proved. The inequality is used to prove existence and uniqueness of solutions of equations of variation-of-constants type. As a consequence, it is shown that the solution of a semilinear delay differential equation with functional The research by M. Reiß was kindly supported by the Deutsche Forschungsgemeinschaft. O. van Gaans acknowledges the financial support provided through the European Community s Human Potential Programme under contracts HPRN-CT-2-1, DYN- STOCH and HPRN-CT and by a VIDI-subsidie ( of the Netherlands Organisation for Scientific Research (NWO. 1

2 Lipschitz diffusion coefficient and driven by a general semimartingale satisfies a variation-of-constants formula. key words: Émery s inequality, functional Lipschitz coefficient, linear drift, semimartingale, stochastic delay differential equation, variation-of-constants formula 2 Mathematics Subject Classifications: 6H2, 6G48, 34K5 1 Introduction The variation-of-constants formula is a key tool in the study of long term behavior of semilinear stochastic differential equations. It describes the diffusion and nonlinearity in the drift as perturbations of a deterministic linear equation and thus enables to reveal information on the long term behavior (see for instance [1, 12]. In the case of stochastic delay differential equations driven by semimartingales, however, such a variation-of-constants formula seemed to be unknown. We will prove in this paper a variation-of-constants formula for stochastic delay differential equations with linear drift and a functional Lipschitz diffusion coefficient driven by a general semimartingale. Our proof includes the extension of several other important results. In particular, we present an extension of Émery s inequality for stochastic integrals. Consider the stochastic delay differential equation dx(t = X(t + aµ(da dt + F (X(t dz(t, t, (,] (1 X(t = Φ(t, t, where µ is a finite signed Borel measure on (, ], Z is a semimartingale, F is a functional Lipschitz coefficient, and (Φ(t t is a given suitable initial process. We want to show that the solution of (1 satisfies the variation-ofconstants formula X(t = g(t s dj(s + g(t sf (X(s dz(s, t, (2 where the process J of bounded variation given by J(t = Φ( + Φ(s + aµ(da ds (3 (, s] 2

3 contains the initial condition. The function g is the fundamental solution of the underlying deterministic delay equation, that is, g [, = g( + aµ(da Lebesgue a.e. on [,, (,] (4 g( = 1, g(t =, t <. It is well known (see [6] that equation (4 has indeed a unique solution g with the property that g [, is absolutely continuous. If a solution of (2 exists, it can be shown by a Fubini argument that it also satisfies the original stochastic delay differential equation (1. Since (1 is known to admit a unique solution, we infer that this solution then satisfies (2. Thus, it remains to prove existence of solutions of (2. Our proof of existence (and uniqueness of solutions of (2 is an extension of the proof for stochastic differential equations presented in [11]. The idea there is to use localization in order to reduce to a Banach fixed point argument in a suitable space of stochastic processes. The key estimate to obtain a contraction is an inequality due to Émery, see [5] or [11, Theorem V.3]. It says that for an adapted càdlàg process Y and a semimartingale Z the size of the stochastic integral can be estimated by Y (s dz(s Y S p Z H q, (5 H r for certain suitable norms on spaces of processes and semimartingales. It turns out that for the more general equations of variation-of-constants type an extension of Émery s inequality is needed, namely for integral processes of the form g( sy (s dz(s, where g is a deterministic function. We will show that for a large class of functions g the inequality g( sy (s dz(s R Y S p Z H q (6 H r holds, where R is a constant independent of Y and Z. We establish an even more general inequality for integrals of the form Y (, s dz(s, where Y belongs to a class of processes with two parameters. With the inequality (6 we can prove that the variation-of-constants type equation X(t = J(t + g(t sf (X(s dz(s, t, (7 3

4 has a unique (up to indistinguishability adapted càdlàg solution X, for any semimartingales J and Z. The nonlinear coefficient F is here assumed to be functional Lipschitz. With the aid of the solution of (7 we are then able to prove the next variation-of-constants formula for stochastic delay differential equations. For abbreviation, we denote by D the space of all adapted càdlàg processes on a filtered probability space that satisfies the usual conditions. Theorem 1.1. Let µ be a finite signed Borel measure on (, ] and let g : R R be the unique solution of (4 with g [, absolutely continuous. Let F : D D be functional Lipschitz. Let J and Z be semimartingales. Then a process X D satisfies X(t = J(t + X(s + aµ(da ds (8 + ( s,] F (X(s dz(s, t, if and only if it satisfies the variation-of-constants formula (2. Moreover, there exists one and only one X D satisfying (8 and (2. The outline of the paper is as follows. In order to remain self-contained, Section 2 settles notation and briefly reviews the basic constructions and tools that we need in the sequel. In Section 3 we prove an inequality of Emery type for stochastic integrals of two parameter processes. Section 4 then derives the inequality (6. Existence and uniqueness of solutions of equation (7 are discussed in Section 5. Finally in Section 6 we prove the variation-of-constants formula of Theorem Preliminaries 2.1 Processes All random variables and stochastic processes are assumed to be defined on a fixed filtered probability space (Ω, F, (F t t, P, where the filtration (F t t satisfies the usual conditions (see [9, Definition ]. Let I [, be an interval and let D(I denote the set of all adapted processes (X(t t I with paths that are almost surely càdlàg (that is, right-continuous and the left limit exists at every t I distinct from the left endpoint of I. If X, Y D(I 4

5 satisfy X(t = Y (t a.s. for every t I, then they are indistinguishable, that is, X(t = Y (t for all t I a.s. (see [9, Problem 1.1.5]. We will identify processes that are indistinguishable. Every process X D(I is jointly measurable from I Ω R (see [9, Remark ]. For a process X D(I, where I, and a stopping time T we define the stopped process X T by (X T (t(ω = X(t T (ω(ω, ω Ω, t I, and X T by X(t(ω1 { t<t (ω} + X(t T (ω (ω1 {t T (ω} (X T (t(ω = if T (ω >, if T (ω =, ω Ω, t I. Here (X(t T (ω (ω = lim s T (ω X(t s(ω for ω Ω with T (ω >. Stopping times are allowed to attain the value. The jumps of a process X D(I, where I [, is an interval with left endpoint a, are defined by ( X(t = X(t X(t for t I, t a, and ( X(a = X(a. Further, by convention, X(a =. For an interval I and a function f : I R we define the total variation of f over I as m 1 Var I (f = sup f(t k+1 f(t k k= where the supremum is taken over all t,..., t m I with t t 1 t m and all m N. A process X D(I with paths that have almost surely finite total variation over each bounded subinterval of I will be called an FV-process and Var I (X is defined pathwise. Let L p denote the Lebesgue space L p (Ω, F, P, where 1 p. For a process (X(t t I for which sup t I X(t is F-measurable, define (possibly and X S p (I = sup t I X(t L p S p (I = {X D(I : X S p (I < }. Notice that for X D(I indeed sup t I X(t equals sup t I Q X(t a.s., which is F-measurable. More than that, if a process (X(t t I is only jointly 5

6 measurable from I Ω to R then the pathwise supremum sup t I X(t is already F-measurable, as follows from [2, III.13 and III.33 or III.44], since F is P-complete. If the interval of definition is clear from the context, we will simply write S p and S p. 2.2 Semimartingales We adopt the definitions and notation of [3, 7, 8]. Recall that a process X D[, is called a semimartingale if there exist a local martingale M and an FV-process A such that X(t = X( + M(t + A(t a.s. for all t. For two semimartingales X and Y we denote by [X, Y ] their covariation (see [3, VII.42] or [8, p. 519]. For any semimartingale X the process [X, X] is positive and increasing (see [8, Theorem 26.6(ii]. We denote [X, X] = sup t [X, X](t. We will use the above terminology also for processes X D[a, b], where a b. We say that X D[a, b] is a local martingale (or semimartingale if there exists a local martingale (or semimartingale Y D[, such that X(t = Y (t for all t [a, b]. If X 1, X 2 D[a, b] are semimartingales and Y 1, Y 2 D[, are semimartingales such that X i (t = Y i (t for all t [a, b], i = 1, 2, then we define [X 1, X 2 ](t := [Y1 b Y1 a, Y2 b Y2 a ](t, t [a, b]. For a semimartingale Z D[a, b] with Z(a = we define Z H p [a,b] = inf{ [M, M](b 1/2 + Var [a,b] (A L p : Z = M + A with (9 (possibly and let M a local martingale, A an FV-process, and M(a = A(a = } H p [a, b] := {Z semimartingale : Z(a =, Z H p [a,b] < }. The space H p [, is defined similarly by replacing the norm in (9 by [M, M] 1/2 + Var [, (A L p. Observe that for any stopping time T and any Z H p [a, b] we have Z T H p [a, b] and Z T H p Z H p. Theorem 2.1. Let 1 p < and a b. The spaces (S p [a, b], S p [a,b] and (H p [a, b], H p [a,b] are Banach spaces. Moreover, if Z H p [a, b] then Z S p [a, b] and there exists a constant c p > (independent of a and b such that Z S p [a,b] c p Z H p [a,b] for all Z H p [a, b]. 6

7 Proof. It is said in [11, p ] that S p [, and H p [, are norms. It is straightforward that (S p [,, S p [, is complete. Completeness of H p [, endowed with H p [, is mentioned in [3, VII.98(e]. The sets {X S p [, : X b = X, X a = } and {X H p [, : X b = X, X a = } are closed subspaces of S p [, and H p [,, respectively, and they are isometrically isomorphic to S p [a, b] and H p [a, b]. The existence of c p is the content of [11, Theorem V.2]. The next statement easily follows from [11, Theorem V.1, Corollary, p ]. Corollary 2.2. Let 1 p and let a b. If Z H p [a, b], then [Z, Z](b 1/2 L p and [Z, Z](b 1/2 L p Z H p [a,b]. Further, if M D[a, b] is a local martingale with M(a = and [M, M](b 1/2 L p, then M H p [a, b] and 2.3 Stochastic integrals M H p [a,b] = [M, M](b 1/2 L p. We use the stochastic integral as presented in [3, 7, 8]. Let us summarize the properties which we need. The predictable σ-algebra P is the σ-algebra in [, Ω generated by the processes (X(t t that are adapted to (F t t and which have paths that are left-continuous on (,. Here F t is the σ-algebra generated by F s with s < t if t >, and F := F. A process X is predictable if (t, ω X(t, ω is measurable with respect to the predictable σ-algebra. Then X is jointly measurable from I Ω to R. For an interval I containing, a process (X(t t I is locally bounded if there exist stopping times T k, that is, = T T 1... with sup k T k = a.s., such that for each k there is a constant c k with, a.s., X T k (t X( ck for all t I. For any process X D[, the process t X(t is both predictable and locally bounded. We consider the class E of processes of the form H(t = H 1 1 {} (t + H 1 (,t1 ](t + + H n 1 1 (tn 1, (t, t, (1 where H 1, H are F -measurable and H i are F ti -measurable random variables for i 1 such that essup H i < and where = t t 1 7

8 t n =, n N. For a semimartingale X and a process H given by (1 the stochastic integral is defined by (H X(t = H(s dx(s := n i=1 H i 1 (X(t i t X(t i 1 t, t. The next theorem (see [3, VIII.3 and 9], [7, Theorem I.4.31 and I ], or [8, Theorem 26.4], and [8, Theorem 26.6(ii and (v] extends the stochastic integral to all locally bounded predictable processes. Theorem 2.3. Let X be a semimartingale. The map H H X on E has a unique linear extension (also denoted by H H X on the space of all predictable locally bounded processes into the space of adapted càdlàg processes such that if (H n n is a sequence of predictable processes with H n (t K(t for all t, n N and for some locally bounded predictable process K and H n (t(ω H(t(ω for all t, ω Ω and for some process H, then (H n X(t (H X(t in probability for all t. Moreover, for every locally bounded predictable processes H and K the following statements hold: (a H X is a semimartingale; (b K (H X and (KH X are indistinguishable; (c (H X and H X are indistinguishable and (H X( = (H X( = H(X(; (d if X is a local martingale then H X is a local martingale and [H X, H X](t = H(s 2 d[x, X](s for all t ; (e if X is of bounded variation then H X is of bounded variation and Var [, (H X sup H(t Var [, (X; t (f if T is a stopping time, then 1 [,T ] X = X T and (H X T = (H1 [,T ] X = H X T up to indistinguishability. 8

9 It follows that the stochastic integral H(s dx(s, t, is well defined for any H D[, and any semimartingale X. If X is an FVprocess and H D[,, then the stochastic integral H(s dx(s equals the pathwise defined Stieltjes integral. The precise formulation of Émery s inequality reads as follows (see [5] or [11, Theorem V.3]. Theorem 2.4 (Émery s inequality. Let p, q, r [1, ] be such that 1 + p 1 = 1 (with the convention that 1 =. Let T >. For every process q r (Y (t t [,T ] in S p [, T ] and every semimartingale (Z(t t [,T ] in H q [, T ] the process ( Y (s dz(s t [,T ] is in H r [, T ] and Y (s dz(s Y S p [,T ] Z H q [,T ]. H r [,T ] In Section 3 and Section 6 we need the following stochastic Fubini theorem. Theorem 2.5. Let (A, A be a measurable space and let µ be a finite signed measure on A. Let Φ: A [, Ω R be an A P-measurable map, where P denotes the predictable σ-algebra in [, Ω. Let Z be a semimartingale with Z( =. If for each a A the process Φ(a, is locally bounded, then (a for every a A there exists an adapted càdlàg version (I a (t t of the stochastic integral ( Φ(a, s dz(s t such that the map (a, t, ω I a (t, ω is A B([, F-measurable; (b if moreover the process Φ(a, A 2 µ (da is locally bounded, then a.s. ( ( Φ(a, s dz(s µ(da = Φ(a, sµ(da dz(s, t, A A where for the inner integral at the left hand side the versions of (a are chosen. Proof. Due to [11, Theorem IV.15], [11, Corollary IV.44] yields (a. Observe that the measurability conditions provide that the process A Φ(a, 2 µ (da is predictable. Assertion (b follows therefore by linearity from [11, Theorem IV.46], again due to [11, Theorem IV.15]. 9

10 3 A class of processes with two parameters In this section we prove the next theorem. Theorem 3.1. Let 1 p, 1 q, and 1 r < be such that = 1. Let T > and let (Y (t, s p q r t,s [,T ] be a process such that Y (t, is an adapted càdlàg process for every t [, T ] and such that there exists a process (Y 1 (t, s t,s [,T ] with Y (t, s = Y (, s + Y 1 (u, s du for all t [, T ] a.s. for each s [, T ], where (t, s, ω Y 1 (t, s, ω is B([, T ] P- measurable and such that Y 1(u, 2 du is a locally bounded process and sup u [,T ] Y 1 (u, S p <. Then for every Z H q [, T ] the process Y (, s dz(s has a version which is a semimartingale and for this version we have Y (, s dz(s Γ p (Y Z H q [,T ], H r [,T ] where Γ p (Y = Y (, S p [,T ] + (1 + c r T sup Y 1 (t, S p. (11 t [,T ] If Γ p (Y <, then Y (, s dz(s Hr [, T ]. We use Y (, as shorthand notation for the process (Y (t, t t [,T ]. Similarly, Y (, s dz(s denotes ( Y (t, s dz(s t [,T ], where for each t the random variable Y (t, s dz(s is a priori only defined up to a set of probability zero. As soon as we have shown that it has a semimartingale version, we will always choose that version. Notice that for each u the process Y 1 (u, is jointly measurable from [, T ] Ω to R, so that sup t [,T ] Y 1 (u, t : Ω [, ] is F-measurable. Throughout the section let p, q, r, T, and the processes (Y (t, s t,s [,T ] and (Y 1 (t, s t,s [,T ] be as in Theorem 3.1. Let further Z D[, T ] be a semimartingale with Z = M + A, M( = A( =, where M is a local martingale with [M, M](T 1/2 L q and A an FV-process with Var [,T ] (A L q. We divide the proof of Theorem 3.1 into several lemmas. We will consider the integral process Y (, s dz(s, substitute Y 1, and apply the stochastic Fubini theorem. We begin by estimating the quadratic variation and total variation of the ensuing terms. 1

11 Lemma 3.2. The map s Y (s, s is a predictable locally bounded process and [ ] Y (s, s dm(s, Y (s, s dm(s (T sup Y (s, s 2 [M, M](T. s [,T ] Proof. As (s, ω Y (, s, ω is P-measurable and (t, s, ω Y 1 (t, s, ω is B([, T ] P-measurable, we have that (t, s, ω Y (t, s, ω is B([, T ] P-measurable. Hence (s, ω Y (s, s, ω is P-measurable. Further, the process (Y (, s s [,T ] is adapted and càdlàg, so that Y (, is locally bounded, and ( s Y 1(u, s du s [,T ] is locally bounded since ( Y 1(u, s 2 du s [,T ] is locally bounded. Hence (Y (s, s s [,T ] is locally bounded. The inequality follows from Theorem 2.3(d. Lemma 3.3. The process Y (, s da(s is an FV-process and Var [,T ] Y (, s da(s ( sup s [,T ] Y (s, s + Var [s,t ] (Y (, s Var [,T ] (A. Proof. If we consider the stochastic integrals as pathwise Stieltjes integrals we find for a partition = t t 1 t n = T a.s. m 1 ( 1[,tk+1](sY (t k+1, s 1 [,tk ](sy (t k, s da(s As k= ( Var [,T ] (A sup Var [,T ] 1[, ] (sy (, s. s [,T ] ( Var [,T ] (1 [, ] (sy (, s Y (s, s + Var [s,t ] Y (, s, we obtain the desired inequality. Since ( Y 1(u, s 2 du s [,T ] is a locally bounded process, it has a.s. bounded paths and since Y (, is càdlàg, it follows that sup s [,T ] Y (s, s < a.s. It remains to show that Y (, s da(s is a càdlàg process, which comes down to showing that U(, s da(s has a.s. càdlàg paths, where U(t, s = Y 1(u, s du, t, s [, T ]. For any n N, set t n k := kt/n and n 1 V n (t, s := 1 [t n k,t n k+1 (tu(t k, s, s, t [, T ], k= 11

12 and observe that a.s. we have sup U(t, s da(s t [,T ] V n (t, s da(s Var [,T ] (A sup sup sup U(t, s U(t n k, s k t [t n k,tn k+1 s [,T ] Var [,T ] (A ( 1/2 T/n sup Y 1 (u, s du 2, s [,T ] due to the Cauchy-Schwarz inequality. Moreover, the process V n(, s da(s has a.s. càdlàg paths. We finish the proof by uniform approximation of almost every path. By assumption, Y 1(u, 2 du is locally bounded, so there are stopping times T l a.s. such that Y 1(u, s T l 2 du M l on Ω for all s, for certain constants M l, l 1. Thus, for almost every ω, t V n(t, s da(s(ω is càdlàg for every n, there is an l such that T l (ω T for all l l, and sup U(t, s da(s(ω V n (t, s da(s(ω t [,T ] Var [,T ] (A(, ω(m l T/n 1/2 for all n. It follows that U(, s da(s is a càdlàg process and therefore Y (, s da(s is an FV-process. In the next lemma we need Émery s inequality for integrands that are not càdlàg. It is easy to verify that Émery s proof in [5] establishes the inequality V (s dz(s V S p Z H q [,T ]. (12 H r [,T ] for any predictable process (V (t t [,T ] with V S p < and any Z H q. Lemma 3.4. For a suitable version of ( u Y 1(u, s dm(s u [,T ], the process ( u Y 1(u, s dm(s du is an FV-process with ( u L Var [,T ] Y 1 (u, s dm(s du c r T sup Y 1 (u, S p M H q. r u [,T ] 12

13 Proof. Due to the stochastic Fubini Theorem 2.5, there exists for each u [, T ] a version of ( Y 1(u, s dm(s t such that (u, t, ω Y 1(u, s dm(s(ω is B([, T ] 2 F-measurable. Then (u, ω u Y 1(u, s dm(s(ω is B([, T ] F-measurable and this process is adapted as well. Further, ( u r 1/r Y 1 (u, s dm(s(ω P( dω du Ω u = Y 1 (u, s dm(s du Y 1 (u, s dm(s du L r S r c r Y 1 (u, s dm(s du c r T sup Y 1 (u, S p M H q, H r u [,T ] where we have used Émery s inequality (12. It follows firstly that u u Y 1(u, s dm(s is an element of the Bochner space L 1 ([, T ], L r. With the aid of a well known inequality from the theory of Bochner integration (see [4, Lemma III.11.16(b and Theorem III.2.2] we obtain u L T Y 1 (u, s dm(s du u Y 1 (u, s dm(s du. r Secondly, by Fubini s theorem, we infer that u u Y 1(u, s dm(s is a.s. integrable, so that ( u Y 1(u, s dm(s du t has a.s. absolutely continuous paths. For absolutely continuous functions the total variation is given by the L 1 -norm of the weak derivative and thus the assertions follow. Lemma 3.5. The process Y (, s dz(s has a version which is a semimartingale and for that choice, Y (, s dz(s = Y (s, s dm(s u + Y (, s da(s + Y 1 (u, s dm(s du, where the process inside the first pair of parentheses is a local martingale and the process inside the second pair of parentheses an FV-process. Proof. We use the stochastic Fubini Theorem 2.5. Let t [, T ]. The map (a, s, ω 1 [s,t ] (ay 1 (a, s is B([, T ] P-measurable, so 13 L r

14 1 [,T ](ay 1 (a, 2 da is predictable. The latter process is also locally bounded, as 1(a, 2 da is locally bounded. Hence we obtain a.s. ( ϑ 1 [s,t ] (ay 1 (a, s dm(s da ϑ ( = 1 [s,t ] (ay 1 (a, s da dm(s for all ϑ [, T ]. For ϑ = t the right hand side equals Y (t, s dm(s Y (s, s dm(s. It follows that the process at the right hand side of the desired identity is a version of the process at the left hand side. Lemma 3.2, Theorem 2.3(d, and Lemmas 3.3 and 3.4 complete the proof. Proposition 3.6. We have Y (, s dz(s Γ p (Y [M, M](T 1/2 + Var [,T ] (A L q, H r [,T ] where Γ p (Y [, ] is given by (11. Proof. First observe that sup Y 1 (t, s dt s [,T ] L p sup σ [,T ] Y 1 (t, σ dt L p Y 1 (t, S p dt, which is clear for p = and similar to the according step in the proof of Lemma 3.4 for p <. Hence sup Var [,T ] (Y (, s = sup Y 1 (t, s dt T sup Y 1 (t, S p. t [,T ] s [,T ] L p s [,T ] Next, Lemmas together with Hölder s inequality and Corollary 2.2 L p 14

15 yield Y (, s dz(s H r [,T ] [ 1/2 Y (s, s dm(s, Y (s, s dm(s] (T ( u + Var [,T ] Y (, s da(s + Y 1 (u, s dm(s du Γ p (Y [M, M](T 1/2 + Var [,T ] (A L q. (, ] L r Finally, Theorem 3.1 follows from Proposition 3.6 by taking the infimum over the semimartingale representations Z = M + A. 4 Application to convolutions This section concerns an Émery inequality for convolutions of the form g(t sy (s dz(s, t [, T ], where g is a deterministic function, Y an adapted càdlàg process, and Z a semimartingale. If the function g is right-continuous, then the integral process is defined up to versions. If one of them is a semimartingale, we will simply say that the process is a semimartingale. Let W 1, [a, b] denote the space of absolutely continuous functions h from the interval [a, b] into R whose weak derivative h is in L [a, b]. Let further the vector space of pure jump functions of bounded variation P JBV [a, b] consist of all j : [a, b] R such that j(t = α i 1 [ti,b](t, t [a, b], (13 i=1 for some t i [a, b] and α i R, i N, with i=1 α i <. 15

16 Theorem 4.1. Let 1 p, 1 q, and 1 r < be such that = 1. Let T >. If g : [, T ] R is such that g = h + j with p q r h W 1, [, T ] and j P JBV [, T ], then there exists a constant R such that for every Y S p [, T ] and every Z H q [, T ] we have g( sy (s dz(s H r [, T ] and g( sy (s dz(s R Y S p [,T ] Z H q [,T ]. H r [,T ] If j is given by (13, then we have R = h( + (1 + c r T h + α i, i=1 where c r is the constant of Theorem 2.1. The proof is divided into the next two lemmas. We will first study absolutely continuous functions g by means of Theorem 3.1 and then consider pure jump functions. We write H r as shorthand for H r [, T ], S p for S p [, T ], etc. Lemma 4.2. Let 1 p, 1 q, and 1 r < be such that = 1. Let T >. If g : [, T ] R is absolutely continuous with p q r derivative g L [, T ], then for every Y S p and every Z H q one has g( sy (s dz(s Hr and g( sy (s dz(s H r ( g( + (1 + c r T g Y S p Z H q. Proof. We begin by extending g by setting g(t := g( for t (,. Then g is absolutely continuous on (, T ], g (t = for t <, and the supremum norms of g and g are not changed by the extension. We choose a version of g L [, T ] such that the essential supremum g equals sup t [,T ] g (t. Define for s, t [, T ], Y (t, s := g(t sy (s and Y 1 (t, s := g (t sy (s. 16

17 Since g is continuous, Y (t, is an adapted càdlàg process for every t [, T ]. Further, Y 1 (, s L [, T ] a.s. and for t [, T ], Y 1 (u, s du = g (u sy (s du = Y (t, s Y (, s. Also, (t, s, ω Y 1 (t, s, ω is B([, T ] P-measurable, Y 1(u, 2 du = g (u 2 duy ( 2 is a locally bounded process, sup s Y 1 (u, s L p for all u, and sup Y 1 (u, S p g Y S p <. u [,T ] Moreover, Γ p (Y ( g( + (1 + c r T g Y S p <. Hence, an application of Theorem 3.1 completes the proof. The next lemma concerns pure jump functions. Lemma 4.3. Let 1 p, 1 q, and 1 r < be such that = 1. Let T > and let g : [, T ] R be given by p q r g(t = α i 1 [ti,t ](t, t [, T ], i=1 where t i [, T ] and α i R, i N, are such that i=1 α i <. Then for every Y S p and every Z H q one has g( sy (s dz(s Hr and ( g( sy (s dz(s α i Y S p Z H q. H r Proof. Let Y S p and Z H q. Observe that g is a càdlàg function. For m N let m g m (t := α i 1 [ti,t ](t, t [, T ]. i=1 Émery s inequality (Theorem 2.4 yields m ( ti g m ( sy (s dz(s + α i Y (s dz(s H r i=1 m α i Y S p Z H q. 17 i=1 i=1 H r

18 If we apply for n > m the previous inequality to g n g m instead of g m, we obtain that (g n ( sy (s dz(s n is a Cauchy sequence in H r. As H r is complete (Theorem 2.1, it follows that g n( sy (s dz(s converges in H r to some H H r. For fixed t [, T ] we have (g m (t s g(t sy (s g m g Y (s for s t and g m g as m. Hence by Theorem 2.3, s g m (t uy (u dz(u s g(t uy (u dz(u in probability for all s [, t], and in particular for s = t. It follows that H(t = g(t uy (u dz(u and that the desired inequality holds. 5 Existence for equations of variation-ofconstants type In this section we will exploit the extended Émery inequality to show existence and uniqueness for stochastic equations of variation-of-constants type. We follow the proof of existence and uniqueness for stochastic differential equations as given in [11]. Definition 5.1. Let I [, be an interval. A map F : D(I D(I is called functional Lipschitz if there exists an increasing jointly measurable (finite process (K(t t I such that for all X, Y D(I, (a X T = Y T = F (X T = F (Y T for every stopping time T, (b F (X(t F (Y (t K(t sup s I [,t] X(s Y (s a.s. for all t I. Recall that equalities of processes such as in (a are meant up to indistinguishability. It is contained in (b that a functionally Lipschitz map F is well-defined this way. Indeed, if X and Y in D(I are indistinguishable, then (b yields that F (X(t = F (Y (t a.s. for all t I and hence F (X and F (Y are indistinguishable. We will establish the next result by a sequence of lemmas. 18

19 Theorem 5.2. Let (Z(t t be a semimartingale, let J D, let F : D D be a functional Lipschitz map, and let g : [, R be such that g [,T ] = h + j with h W 1, [, T ] and j P JBV [, T ] for every T >. Then the equation X(t = J(t + g(t sf (X(s dz(s, t, has a unique (up to indistinguishability solution X D. If J is a semimartingale, then so is X. Given constants 1 p <, t >, and R >, we will use the following property of a function g : [, t ] R: g is càdlàg and for every Y S p [, t ] and Z H [, t ] we have g( sy (s dz(s Hp [, t ] and g( sy (s dz(s H p [,t ] R Y S p [,t ] Z H [,t ]. (14 Lemma 5.3. Let 1 p <, let t >, and let J S p [, t ]. Let F : D[, t ] D[, t ] be functional Lipschitz as in Definition 5.1 with F ( = and sup t [,t ] K(t k a.s. for some constant k. Let g : [, t ] R be a function and R > be a constant such that (14 is satisfied. Let Z H [, t ] be such that Z H [,t ] 1/2γ, where γ = c p kr. Let T be a stopping time. Then the equation T X(t = J T (t + g( sf (X(s dz(s (t, t t, has a unique solution X in S p [, t ] and X S p [,t ] 2 J S p [,t ]. Proof. Define Λ(X(t := J(t + g(t sf (X(s dz(s, t [, t ], X S p [, t ]. By the assumption (14, the assumption F ( =, and the fact that H p S p we have that Λ(X S p for every X S p. Further, for X, Y S p we have Λ(X Λ(Y = ( g( s F (X(s F (Y (s dz(s H p. 19

20 Moreover, due to assumption (14, Λ(X T Λ(Y T H p Λ(X Λ(Y H p R F (X F (Y S p Z H 1 2c p k k X Y S p, so Λ(X T Λ(Y T S p c p Λ(X T Λ(Y T H p 1 X Y 2 Sp. Since S p is complete, we find a unique fixed point X S p of Λ( T. This X is the solution as asserted, and X J S p = Λ(X T Λ( T S p 1 2 X S p, so that X S p X J S p + J S p 1 X 2 S p + J S p 2 J S p. and hence X S p Definition 5.4. (see [11, p.192] Let I = [, or I = [a, b] for some a b. Let Z H (I, and α >. Then Z is called α-sliceable, denoted by Z S(α, if there exist stopping times = T T 1 T k such that Z = Z T k and (Z Z T i T i+1 H (I α for i =,..., k 1. Theorem 5.5. (see [11, Theorem V.5] Let Z D[, be a semimartingale with Z( = a.s. (a If α >, Z S(α, and T is a stopping time, then Z T Z T S(α. S(α and (b For every α > there exist stopping times T k, that is, = T T 1 and sup k T k = a.s., such that Z T k S(α for all k. It follows that for every α >, t >, and Z H [, t ] there exist stopping times T k such that Z T k S(α for all k. The next lemma extends Lemma 5.3 to more general semimartingales. Lemma 5.6. The existence and uniqueness assertions of Lemma 5.3 remain true if the condition Z H 1/(2γ is relaxed to Z S(1/(2γ. Proof. Let S, S 1,..., S l be stopping times such that = S S 1 S l, Z = Z Sl, and (Z Z S i Si+1 H 1/2γ for i l 1 (these exist because Z S(1/2γ. Let T i := S i T, i =,..., l. Then = T T 1 T l and (Z Z T i Ti+1 H = ((Z Z S i Si+1 T H 1/2γ. We 2

21 argue by induction on i. If the equation stopped at T i has a unique solution, we first show that the equation stopped at T i has a unique solution and then we show existence and uniqueness for the equation stopped at T i+1. Suppose that i {,..., l 1} is such that the equation X(t = J Ti (t + Ti g( sf (X(s dz Ti (s (t, t t, (15 has a unique solution X S p. In order to simplify notation, we extend F (U(t := F (U(t, J(t := J(t, and Z(t := Z(t for t t and U D[, t ]. Further we interpret [c, b] = if c > b. Let ( ( Y := X + J(T i + g( sf (X(s dz(s (T i 1 [Ti,t ]. Observe that Y T i = X T i and Y T i = Y. Next we claim that Y is the unique solution in S p of Ti V (t = J T i (t + g( sf (V (s dz T i (s (t, t t. (16 For a proof, notice that we have Y Ti = X Ti, so F (Y Ti = F (X T i and Ti J T i (t + g( sf (Y (s dz(s (t = X(t + J(T i 1 [Ti,t ](t + g( sf (X(s dz(s (T i 1 [Ti,t ](t = Y (t, t t, so Y satisfies (16. Further, sup Y (s X S p + J S p + 2R F (X S p Z H, s [,t ] L p so Y S p. To see uniqueness, suppose that V S p is another solution of (16. Then V T i satisfies the equation for X, so V T i = X T i = Y T i. From 21

22 (16 it is clear that V = V T i and that ( V T i V Ti = J(T i + g( sf (X(s dz(s (T i 1 [Ti,t ] = Y T i Y T i, so that V = Y. Thus the claim has been proved. Let us introduce D i U := (U U T i Ti+1 and G(U = F (Y + U F (Y, U D[, t ]. Consider the equation ( U(t = D i J(t + Ti+1 g( sf (Y (s dz Ti+1 (s (t (17 g( sf (Y (s dz(s + Ti T i+1 (t Ti+1 g( sg(u(s dd i Z(s (t, t t, We claim that (a equation (17 has a unique solution U in S p, and (b the process V := U + Y Ti+1 is the unique solution of Ti+1 V (t = J Ti+1 (t+ g( sf (V (s dz Ti+1 (s (t, t t. For a proof of (a, observe that the sum of the first three terms of (17 is a member of S p, G is functional Lipschitz with G( = and satisfying the same estimates as F, and D i Z(s H = (Z Z T i T i+1 H < 1/2γ. Now apply Lemma 5.3. To show (b, notice that it is clear from (17 that U = U Ti+1 and U Ti =. Consequently, F (Y + U Ti = F (Y Ti. Due to (16 we can easily check that Ti+1 U + Y Ti+1 = J Ti+1 + g( sf (Y + U(s dz Ti+1 (s. For any solution V, the process V Y T i+1 satisfies (17 and therefore equals U. Thus we have proved the claim. 22

23 We conclude that if equation (15 has a unique solution X in S p, then the equation (15 with T i replaced by T i+1 has a unique solution in S p as well. As for i =, X = is the unique solution X in S p, we find that there exists a unique solution V S p of (15 with i = l. Finally, let T X(t := J T (t + g( sf (V (s dz(s (t, t [, t ]. Because T l = T S l and Z Sl = Z we have X Tl = V and hence T g( sf (X(s dz(s = g( sf (X Tl (s dz Tl (s T = g( sf (V (s dz(s = X J T. T We will increase the generality of the assumptions building on Lemma 5.6 in Proposition 5.8 below. The next lemma is needed in the proof of Proposition 5.8. Lemma 5.7. Let Y D[, and let Z D[, be a semimartingale. Let 1 p < and let g : [, R be a function such that for every t > there exists a constant R > such that (14 is satisfied. Then ( g(t sy (s dz(s is a semimartingale. Proof. By convention, Y ( =, so we may assume Z( =. Observe that there exist stopping times T k such that Y T k S p [, for all k. Use Theorem 5.5 to choose the stopping times T k such that also Z T k H [, for each k. Then for each t >, ( g( sy T k (s dz T k (s t [,t ] H p [, t ]. Hence g( sy (s dz(s Tk t = t Tk t g( sy Tk (s dz Tk (s equals a stopped semimartingale. It follows that g( sy (s dz(s is a local semimartingale and hence a semimartingale by [7, Proposition I.4.25(a and (b]. 23

24 Proposition 5.8. Let Z be a semimartingale, let J D[,, and let F : D[, D[, be functional Lipschitz. Let g : [, R be a function such that for every t > there exists a constant R > such that (14 is satisfied. Then X(t = J(t + g(t sf (X(s dz(s, (18 t, has a unique solution X in D[,. If J is a semimartingale, then so is X. Proof. We use the notation of Definition 5.1. As F (X( = for all X, we may assume that Z( =. We begin by replacing J by J + g( sf ((s dz(s and F by F ( F (. Thus we may assume that F ( =. We proceed by showing the following claim. Let t >. Suppose that K(t, ω k for a.e. ω and all t t. Let S be a stopping time. Then there is a unique process X D such that S X(t = J S (t + g( sf (X(s dz(s (t, t t. (19 To prove the claim, let R > be a constant corresponding to t such that (14 is satisfied. Let γ := c p kr. For every stopping time T such that J T S 2 and Z T S(1/2γ, Lemma 5.6 says that there is a unique X T S 2 such that S X T = (J T S + g( sf (X T (s dz (s T. By uniqueness we have for any two such stopping times T 1 and T 2 that X T 3 T 1 = X T 3 T 2, where T 3 = T 1 T 2. Due to Theorem 5.5, there exist stopping times T l such that J Tl S 2 and Z Tl S(1/2γ for all l. Define X(t := X Tl (t1 [Tl 1,T l (t, t t. l=1 Then (X(t t is an adapted càdlàg process and X S = X. Further, for 24

25 l 1, we have X T l = X Tl and by (a of Definition 5.1, ( ( J + S Tl g( sf (X(s dz(s = (J T l S + = X Tl = X T l. ( Tl S g( sf (X Tl (s dz(s It follows that X satisfies (19. To show uniqueness, let Y be another adapted càdlàg solution of (19. There exist stopping times S l with Y Sl S 2 for all l. Then Y (S l T l satisfies the same equation as X S l T l and by uniqueness we obtain Y (S l T l = = X (S l T l. Since sup l S l T l = a.s., it follows that X = Y and we have proved the claim. Next, fix t >. For n N define the set X S l T l Ω n := {ω Ω: K(t, ω n}. Then Ω n F for each n and P( n Ω n = 1. There is an n N such that P(Ω n > for all n n. Define for n n a probablity P n on (Ω, F by P n (A := P(A Ω n P(Ω n 1, A F. Let F n be the P n -completion of F and F n t the P n -completion of F t, for t. The filtration (F n t t in (Ω, F n, P satisfies the usual conditions. Further, Z is a semimartingale on (Ω, F n, (F n t t, P n. Relative to P n we have K(t n a.s. for all t [, t ], so by the first claim there exists an (F n t t -adapted process X n with P n -a.s. càdlàg paths such that X n (t = J(t + g(t sf (X n (s dz(s, t t, (2 P n -a.s. Since for n > n, P n 1 is absolutely continuous with respect to P n, the stochastic integral g(t sf (Xn (s dz(s defined on (Ω, F n 1, P n 1 equals P n 1 -a.s. the stochastic integral with the same notation defined on (Ω, F n, P n (see [11, Theorem II.14]. Hence X n is an (Ft n 1 t -adapted càdlàg process on (Ω, F n 1, P n 1 for which (2 holds P n 1 -a.s. Due to uniqueness, 25

26 we obtain that X n (t = X n 1 (t for all t [, t ] P n 1 -a.s. Define X(t, ω := 1 Ωn (ωx n (t, ω+ n=n +1 1 Ωn\Ω n 1 (ωx n (t, ω, ω Ω, t [, t ]. Then X has P-a.s. càdlàg paths, as 1 Ωn X n ( has P-a.s. càdlàg paths for all n. To see that X is adapted, let t [, t ] and B B(R. For n N, {X n (t B} Ft n, so there exists A n F t with 1 {X n (t B} = 1 An P n - a.s. Then P-a.s., 1 {X(t B} 1 Ωn = 1 {X n (t B}1 Ωn = 1 An 1 Ωn for all n. Let A := l=1 m=l A m F t. Then P-a.s., 1 A 1 Ωn = inf l n sup m l 1 Am 1 Ωn = inf l n sup m l 1 Am 1 Ωm 1 Ωn = 1 {X(t B} 1 Ωn for all n, hence {X(t B} F t. Moreover, X n and X are P n -indistinguishable, so F (X n and F (X are P n - indistinguishable and therefore (18 holds for t t P n -a.s. Thus X is the unique adapted càdlàg process satisfying (18 for all t [, t ] P-a.s. Finally, we can vary t and glue solutions together to obtain a unique X D such that (18 holds for all t. It follows from Lemma 5.7 that X is a semimartingale whenever J is a semimartingale. Theorem 5.2 follows from Theorem 4.1 and Proposition 5.8. Notice that the function g in Theorem 5.2 need not be continuous. In this way Theorem 5.2 generalizes [1]. 6 Variation-of-constants formula for SDDE with linear drift It is the aim of this section to prove Theorem 1.1. It is well known that (4 has a unique solution g : R R with g [, absolutely continuous (see [6]. Then (,] g( +aµ(da is bounded on [, T ] and hence g [,T ] W 1, [, T ] for every T >. The proof of Theorem 1.1 proceeds as follows. Due to Theorem 5.2, there exists a solution of (2. By means of a stochastic Fubini argument, we will show that this solution also satisfies (8. As equation (8 has only one solution, we then know that the solutions of (8 and (2 coincide and the proof is complete. The Fubini argument is given next. Lemma 6.1. Let µ be a finite signed Borel measure on (, ] and let g : R R be the solution of (4 with g [, absolutely continuous. Let 26

27 F : D[, D[, be functional Lipschitz. Let (Z(t t and (J(t t be semimartingales. If X D[, satisfies then X(t = g(tx( + + X(t = X( + J(t + + g(t s dj(s (21 g(t sf (X(s dz(s, t, ( s,] X(s + aµ(da ds F (X(s dz(s, t. Proof. Observe that we may assume that Z( = and J( =. We will first apply the stochastic Fubini theorem twice to prove the identity (,] = (s+a + g(s + a mf (X(m dz(mµ(da ds (22 g(t mf (X(m dz(m F (X(m dz(m, for any T and any X D. Since we will only evaluate g on (, T ], we may assume that g is bounded. Fix an s [, T ]. The map (a, t, ω g(s+a tf (X(t (ω is bounded and B((, ] P-measurable. Further, the process F (X( is predictable and locally bounded, the function g is Borel measurable and bounded, and g(s + a 2 F (X( 2 µ (da g 2 µ ((, ]F (X( 2. (,] The stochastic Fubini theorem (Theorem 2.5 therefore yields that ( g(s + a mf (X(m dz(m µ(da (,] = ( g(s + a mµ(daf (X(m (,] 27 dz(m a.s.

28 for every t. Since g(ϑ = for ϑ <, the inner integral at the left hand side of the previous equality runs only up to (s + a + if t s. The map (s, m, ω g(s + a mµ(daf (X(m is measurable with respect (,] to B([, T ] P, because (s, m g(s + a mµ(da is B([, T ] (,] P-measurable. Moreover, for each s the processes g(s + a (,] µ(daf (X( and ( (,] g(s + a µ(daf (X( 2 ds are locally bounded, since g is bounded, µ is a finite measure, and F (X( is locally bounded. Hence, again by the stochastic Fubini theorem (Theorem 2.5, we have ( = (,] ( g(s + a mµ(daf (X(m dz(m ds (,] g(s + a mµ(daf (X(m ds dz(m a.s. for every t. Next, we substitute t = T in the previous equality, use that g(ϑ = for ϑ <, and rewrite the right hand side by noting that for m, m (,] g(s + a mµ(da ds = Thus we arrive at the identity (22. Similarly, we have for any T that (,] = (s+a + m g (s ds = g(t m 1. g(s + a m dj(mµ(da ds (23 g(t m dj(m dj(m. Next assume that X D satisfies (21. Set X(t := for t <. Then X(u = g(ux( + u + g(u s dj(s u + + g(u sf (X(s dz(s, for all u R. 28

29 Therefore, ( s,] = = X(s + aµ(da ds = (,] (,] g(s + aµ(da dsx( (s+a + (,] (s+a + (,] g (s dsx( + ( = g(t 1 X( + + X(s + aµ(da ds g(s + a m dj(mµ(da ds g(s + a mf (X(m dz(mµ(da ds g(t m dj(m J(t g(t mf (X(m dz(m g(t m dj(m J(t g(t mf (X(m dz(m = X(t X( J(t F (X(m dz(m, F (X(m dz(m F (X(m dz(m for all t. Here the third equality is justified by the identities (22 and (23, which yields the assertion. Acknowledgment. The authors thank Igor Stojković and the referee for several corrections. References [1] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of mathematics and its applications, Vol. 45, Cambridge University Press, Cambridge, [2] Dellacherie, C.; Meyer, P.-A., Probabilité et potentiel, Chapitres I à IV, Édition entièrement refondue, Hermann, Paris,

30 [3] Dellacherie, C.; Meyer, P.-A., Probabilité et potentiel, Chapitres V à VIII, Théorie des martingales, Édition entièrement refondue, Hermann, Paris, 198. [4] Dunford, N.; Schwartz, J.T., Linear Operators, Part 1: General Theory, Wiley Interscience, New York, [5] Émery, M., Stabilité des solutions des équations différentielles stochastiques applications aux intégrales multiplicatives stochastiques, Z. Wahrscheinlichkeitstheorie verw. Gebiete 41 (1978, [6] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, Applied Mathematical Sciences 99, Springer-Verlag, New York, [7] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, Second edition, Springer-Verlag, Berlin, Heidelberg, 23. [8] Kallenberg, O., Foundations of Modern Probability, Second Edition, Springer-Verlag, New York, 22. [9] Karatzas, I.; Shreve, S.E., Brownian Motion and Stochastic Calculus, Second Ed., Graduate Texts in Mathematics 113, Springer-Verlag, New York, [1] Protter, Ph., Volterra equations driven by semimartingales, Ann. Probab. 13 (1985 no. 2, [11] Protter, Ph., Stochastic Integration and Differential Equations, A New Approach, Springer-Verlag, Berlin, Heidelberg, 199. [12] Reiß, M.; Riedle, M.; van Gaans, O., Delay differential equations driven by Lévy processes: Stationarity and Feller properties, Stochastic Process. Appl. 116 (26, no. 1,