The randomization method in stochastic optimal control

TL;DR

This paper surveys the randomization method in stochastic optimal control, highlighting its ability to represent value functions via BSDEs and its broad applicability to nonlinear PDEs, along with open problems.

Contribution

It provides a comprehensive overview of the randomization method, detailing its theoretical foundations, applications, and open challenges in stochastic control.

Findings

The method effectively represents value functions using BSDEs.

It applies to a wide class of control problems, including fully nonlinear PDEs.

The paper identifies open problems for future research.

Abstract

In this paper we make a survey on the so called randomization method, a recent methodology to study stochastic optimization problems. It allows to represent the value function of an optimal control problem by a suitable backward stochastic differential equation (BSDE), by means of an auxiliary optimization problem having the same value as the starting one. This method works for a large class of control problems and provides a BSDE representation to many related PDEs of Hamilton-Jacobi-Bellman type, even in the fully non linear case. After a general informal introduction we explain the method giving full details in a basic case. Then we try to give a complete picture of the existing applications and we present some related open problems.