Control of Conditional Processes and Fleming--Viot Dynamics
Philipp Jettkant
TL;DR
The paper studies control problems for conditional diffusion processes killed at the boundary, with rewards depending on the survival-conditioned law. It proves that open- and closed-loop formulations are equivalent by combining measurable-selection (Haussmann–Lepeltier) with mimicking (Brunick–Shreve), and it connects the conditional McKean–Vlasov dynamics to Fleming–Viot type dynamics where reinsertion occurs according to the current law. A contraction mapping argument on measure flows yields well-posedness of the conditional MV SDE, and a subsequent FV interpretation shows that the control problem can be reformulated in terms of mean-field reinsertion dynamics, broadening potential applications and linking to prior work on soft killing and particle systems. The results provide a rigorous foundation for the equivalence of control formulations and offer a new perspective on reinsertion costs and mean-field limits in stochastic control.
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.