Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

TL;DR

This paper establishes a novel representation for adapted solutions of backward stochastic Volterra integral equations (BSVIEs) using partial differential equations and stochastic differential equations, enhancing understanding of their regularity and solution structure.

Contribution

It introduces a new representation framework for adapted M-solutions of BSVIEs, linking them to PDEs and SDEs, which was not previously developed.

Findings

Representation of solutions via PDEs and SDEs

Existence and uniqueness of the representation

Regularity properties of the solutions

Abstract

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward-backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.