A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

TL;DR

This paper introduces a new framework for analyzing type-I backward stochastic Volterra integral equations, establishing their well-posedness through an infinite-dimensional system and linking them to PDEs, with applications in stochastic control.

Contribution

It develops a unified approach to well-posedness of type-I backward stochastic Volterra equations by connecting them to infinite-dimensional SDEs and PDE representations, advancing theoretical understanding.

Findings

Established well-posedness of a new class of multidimensional type-I backward stochastic Volterra equations.

Proved equivalence between the Volterra equations and an infinite-dimensional system of backward SDEs.

Applied the framework to time-inconsistent stochastic control, showing equivalence of different approaches.

Abstract

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semilinear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.