Stochastic control with self-exciting processes

TL;DR

This paper develops a stochastic maximum principle framework for optimal control of SDEs driven by self-exciting processes, addressing non-Markovian challenges and deriving key theoretical conditions.

Contribution

It introduces a novel stochastic maximum principle for non-Markovian self-exciting processes and provides explicit expressions and conditions for optimal control.

Findings

Derived a sufficient stochastic maximum principle for self-exciting processes.

Established a necessary maximum principle (equivalence) for the control problem.

Applied the framework to a log-utility optimization example.

Abstract

We analyze the problem of stochastic optimal control of SDEs where the driver includes a self-exciting stochastic process. Due to the non-Markovian nature of the problem, we apply the stochastic maximum principle approach. We derive a sufficient stochastic maximum principle under this framework. We also derive an expression via martingales of both the self-exciting process and its quadratic covariation. Furthermore, we derive a necessary maximum (equivalence principle) for the self-exciting stochastic control problem. Finally, we look at an application to log-utility.