Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems

TL;DR

This paper demonstrates that for bilinear stochastic control systems with slow and fast dynamics, the optimal control and cost can be approximated by reduced models using homogenization and FBSDEs, simplifying computations.

Contribution

It introduces a method to approximate the optimal control of complex bilinear systems via reduced order models using homogenization and FBSDEs, validated by numerical examples.

Findings

Optimal cost converges to an effective cost in the time scale limit.

Reduced order stochastic control approximates full system control effectively.

FBSDEs can be efficiently solved using least squares Monte Carlo methods.

Abstract

We study linear-quadratic stochastic optimal control problems with bilinear state dependence for which the underlying stochastic differential equation (SDE) consists of slow and fast degrees of freedom. We show that, in the same way in which the underlying dynamics can be well approximated by a reduced order effective dynamics in the time scale limit (using classical homogenziation results), the associated optimal expected cost converges in the time scale limit to an effective optimal cost. This entails that we can well approximate the stochastic optimal control for the whole system by the reduced order stochastic optimal control, which is clearly easier to solve because of lower dimensionality. The approach uses an equivalent formulation of the Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs (FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares Monte Carlo algorithm and show its applicability by a suitable numerical example.