Stochastic near-optimal control: additive, multiplicative, non-Markovian and applications

TL;DR

This survey discusses a novel discretization approach for near-optimal stochastic control, called the skeleton structure, enabling epsilon-optimal solutions in complex non-Markovian systems and highlighting open challenges.

Contribution

It introduces the skeleton discretization method for stochastic control, providing a new way to achieve near-optimality in non-Markovian systems beyond traditional time-discretization.

Findings

The skeleton approach yields epsilon-optimal controls in non-Markovian systems.

Illustrative example demonstrates the effectiveness of the technique.

Highlights open problems in geometrical frameworks and discontinuous noise.

Abstract

In this survey we present the near-optimal stochastic control problem according to some recent tools in the literature. In particular, we focus on the approach of a discretization of the noise values instead of the canonical time-discretization. This is the so called {\it skeleton} structure. This allows to obtain an $\epsilon$-optimal control in non-Markovian systems (the main Theorem). A simple example illustrates the technique. The importance of the approach is emphasised in a final section on open problems related to more geometrical framework and discontinuous noise.