Finite Horizon Robust Impulse Control in a Non-Markovian Framework and Related Systems of Reflected BSDEs

TL;DR

This paper addresses a finite horizon robust impulse control problem for non-Markovian stochastic systems, establishing existence and uniqueness of solutions via interconnected reflected BSDEs and Picard iteration.

Contribution

It introduces a novel framework for robust impulse control in non-Markovian settings using systems of reflected BSDEs with stochastic Lipschitz coefficients.

Findings

Existence of solutions to the system of RBSDEs is proven.

Uniqueness of solutions is established through auxiliary impulse control problems.

The approach applies Picard iteration to handle the complex system.

Abstract

We consider a robust impulse control problem in finite horizon where the underlying uncertainty stems from an impulsively and continuously controlled functional stochastic differential equation (FSDE) driven by Brownian motion. We assume that the controller acts upon the system by impulses while the adversary player (nature) acts through continuous controls. We look for a weak solution which leads us to consider a system of sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions to our system of RBSDEs by applying a Picard iteration approach. Uniqueness then follows by relating the limit to an auxiliary impulse control problem.