On the support of solutions to stochastic differential equations with path-dependent coefficients

Abstract

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron-Martin space under the flow of the solutions of a system of path-dependent (ordinary) differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on the Functional Ito calculus. MSC2010 classification: 60H10 ; 28C20 ; 34K50.

Support characterization for regular path-dependent stochastic Volterra integral equations

“…then the converse inclusion holds. The sufficiency of (2.4) and (2.5) follows from a basic result on the support of probability measures, see [7][Lemma 36] for example. To verify the validity of both limits, we consider a more general setting.…”

“…Then Theorem 1.2 applies and for k b = k σ = 1 it reduces to the support theorem in [7] with the same regularity conditions. Moreover, we could also take…”

“…E-mail: alex.kalinin@mail.de stochastic partial differential equation (SPDE) in Bally et al [3]. By using the vertical derivative as functional space derivative and generalizing the approach in [11] with the relevant Girsanov changes of measures, a path-dependent version of the Stroock-Varadhan support theorem in Hölder norms was recently derived in [7]. The contribution of this article is to extend this support characterization to stochastic Volterra integral equations with regular path-dependent coefficients by providing a flow of mild solutions to ordinary integro-differential equations.…”

“…To be precise, Section 3.1 gives a sufficient condition for a sequence of processes to converge in this sense by exploiting an explicit Kolmogorov-Chentsov estimate. In Section 3.2 we introduce the relevant notations in the context of sequences of partitions and recall a couple of auxiliary moment estimates from [7,10]. The purpose of Section 3.3 is to deduce moment estimates for deterministic and stochastic Volterra integrals, generalizing the bounds from [7][Lemmas 20, 21 and Proposition 22].…”

“…In Section 3.2 we introduce the relevant notations in the context of sequences of partitions and recall a couple of auxiliary moment estimates from [7,10]. The purpose of Section 3.3 is to deduce moment estimates for deterministic and stochastic Volterra integrals, generalizing the bounds from [7][Lemmas 20, 21 and Proposition 22]. Section 4 is devoted to a variety of specific moment estimates and decompositions, preparing the proof of Theorem 2.3.…”