Control of Conditional Processes and Fleming--Viot Dynamics

TL;DR

This paper explores the control of conditional processes, establishing equivalence between open- and closed-loop formulations, and linking the problem to Fleming--Viot dynamics, thus providing new insights and potential applications.

Contribution

It proves the equivalence of open- and closed-loop formulations of the control problem and connects it to Fleming--Viot dynamics, offering a novel interpretation.

Findings

Proved the equivalence of control formulations using measurable selection.

Linked the control problem to Fleming--Viot dynamics with reinsertion.

Provided a new perspective on control with costs due to reinsertion.

Abstract

We discuss equivalent formulations of the control of conditional processes introduced by Lions. In this problem, a controlled diffusion process is killed once it hits the boundary of a given domain and the controller's reward is computed based on the conditional distribution given the process's survival. So far there is no clarity regarding the relationship between the open- and closed-loop formulation of this nonstandard control problem. We provide a short proof of their equivalence using measurable selection and mimicking arguments. In addition, we link the closed-loop formulation to Fleming--Viot dynamics of McKean--Vlasov type, where upon being killed the diffusion process is reinserted into the domain according to the current law of the process itself. This connection offers a new interpretation of the control problem and opens it up to applications that feature costs caused by reinsertion.