Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations

TL;DR

This paper develops a framework for stochastic control and PDE analysis of delayed stochastic differential equations using backward SDEs, with applications to financial markets with memory effects.

Contribution

It introduces a novel approach linking delayed SDEs, backward SDEs, and Hamilton-Jacobi-Bellman equations, providing new tools for control and pricing in markets with memory.

Findings

Characterization of the process Z as a deterministic functional of X

Solution of parabolic PDEs on the space of continuous functions

Application to pricing and hedging in markets with memory

Abstract

We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\mathbf C$, with generator $\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\mathbf C$ driven by $\mathcal L$, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process $X$.