Non-standard Stochastic Control with Nonlinear Feynman-Kac Costs

TL;DR

This paper investigates a conditional stochastic control problem inspired by biology, demonstrating the equivalence of value functions for Markovian and open loop controls, and characterizes solutions via nonlinear PDEs with numerical validation.

Contribution

It provides a rigorous analysis of Lions' conjecture on control value functions, introduces a relaxed formulation with soft killing, and links the problem to deterministic PDE systems.

Findings

Equivalence of value functions for Markovian and open loop controls.

Reduction of stochastic control to deterministic PDE systems.

Numerical experiments validating theoretical results.

Abstract

We consider the conditional control problem introduced by P.L. Lions in his lectures at the Coll\`ege de France in November 2016. In his lectures, Lions emphasized some of the major differences with the analysis of classical stochastic optimal control problems, and in so doing, raised the question of the possible differences between the value functions resulting from optimization over the class of Markovian controls as opposed to the general family of open loop controls. The goal of the paper is to elucidate this quandary and provide elements of response to Lions' original conjecture. First, we justify the mathematical formulation of the conditional control problem by the description of practical model from evolutionary biology. Next, we relax the original formulation by the introduction of \emph{soft} as opposed to hard killing, and using a \emph{mimicking} argument, we reduce the open loop optimization problem to an optimization over a specific class of feedback controls. After proving existence of optimal feedback control functions, we prove a superposition principle allowing us to recast the original stochastic control problems as deterministic control problems for dynamical systems of probability Gibbs measures. Next, we characterize the solutions by forward-backward systems of coupled non-linear Partial Differential Equations (PDEs) very much in the spirit of the Mean Field Game (MFG) systems. From there, we identify a common optimizer, proving the conjecture of equality of the value functions. Finally we illustrate the results by convincing numerical experiments.