Hopf-Lax approximation for value functions of L\'evy optimal control problems

TL;DR

This paper develops a stochastic Hopf-Lax formula approach to approximate value functions in Lévy-driven optimal control problems, providing bounds, convergence rates, and iterative methods for dynamic consistency.

Contribution

It introduces a novel stochastic Hopf-Lax framework for Lévy processes, offering explicit bounds, convergence guarantees, and iterative schemes for value function approximation.

Findings

Provides upper and lower bounds for the value function.

Establishes convergence of the approximation as iterations increase.

Offers explicit rates of convergence and guarantees.

Abstract

In this paper, we investigate stochastic versions of the Hopf-Lax formula which are based on compositions of the Hopf-Lax operator with the transition kernel of a L\'evy process taking values in a separable Banach space. We show that, depending on the order of the composition, one obtains upper and lower bounds for the value function of a stochastic optimal control problem associated to the drift controlled L\'evy dynamics. Dynamic consistency is restored by iterating the resulting operators. Moreover, the value function of the control problem is approximated both from above and below as the number of iterations tends to infinity, and we provide explicit convergence rates and guarantees for the approximation procedure.