Support characterization for regular path-dependent stochastic Volterra integral equations

TL;DR

This paper characterizes the support of solutions to regular path-dependent stochastic Volterra integral equations using functional Itô calculus, revealing the structure of their law in the H"older norm.

Contribution

It introduces a support characterization for solutions of path-dependent stochastic Volterra equations via a flow of integro-differential equations, extending support theory to this class.

Findings

The solution is a semimartingale with H"older continuous paths.

Support of the law is described by a flow of mild solutions to integro-differential equations.

Support characterization leverages the vertical derivative of the diffusion coefficient.

Abstract

We consider a stochastic Volterra integral equation with regular path-dependent coefficients and a Brownian motion as integrator in a multidimensional setting. Under an imposed absolute continuity condition, the unique solution is a semimartingale that admits almost surely H\"older continuous paths. Based on functional It\^o calculus, we prove that the support of its law in the H\"older norm can be described by a flow of mild solutions to ordinary integro-differential equations that are constructed by means of the vertical derivative of the diffusion coefficient.