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

TL;DR

This paper extends the Stroock-Varadhan support theorem to stochastic differential equations with path-dependent coefficients, characterizing the support of solutions using path-dependent ODE flows and functional Ito calculus.

Contribution

It introduces a support theorem for SDEs with path-dependent coefficients, generalizing classical results to a broader class of stochastic processes.

Findings

Support of the solution law is the image of the Cameron-Martin space.

Supports are characterized via flows of path-dependent ODEs.

The approach uses functional Ito calculus.

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 mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of stochastic differential equations with path-dependent coefficients. The proof is based on functional Ito calculus.