Optimal control of conditioned processes with feedback controls

TL;DR

This paper studies finite-horizon stochastic control problems conditioned on a process remaining within a domain, deriving optimality conditions, analyzing asymptotic behavior, and proposing numerical solutions.

Contribution

It introduces a novel framework for conditioned stochastic control problems, including new PDE systems and asymptotic analysis as the horizon extends.

Findings

Derived coupled PDE system with Dirichlet conditions for optimal control

Analyzed asymptotic behavior leading to an eigenvalue problem

Proposed numerical methods validated by simulations

Abstract

We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to $+\infty$. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with numerical simulations.