Deterministic control of SDEs with stochastic drift and multiplicative noise: a variational approach

TL;DR

This paper develops a variational approach to approximate solutions of linear stochastic differential equations with stochastic drift and multiplicative noise, using deterministic controls and Pontryagin's maximum principle.

Contribution

It introduces a novel combination of variational calculus and control theory to identify optimal deterministic approximations for SDEs with stochastic components.

Findings

Necessary and sufficient conditions for optimal controls.

Penalized functionals ensure existence of minimizers.

Expected drift is not always optimal for mean squared error.

Abstract

We consider a linear stochastic differential equation with stochastic drift and multiplicative noise. We study the problem of approximating its solution with the process that solves the equation where the possibly stochastic drift is replaced by a deterministic function. To do this, we use a combination of deterministic Pontryagin's maximum principle approach and direct methods of calculus of variations. We find necessary and sufficient conditions for a function $u \in L^1(0,T)$ to be a minimizer of a certain cost functional. To overcome the problem of the existence of such minimizer, we also consider suitable families of penalized coercive cost functionals. Finally, we consider the important example of the quadratic cost functional, showing that the expected value of the drift component is not always the best choice in the mean squared error approximation.