Mimicking and Conditional Control with Hard Killing

TL;DR

This paper proves a new mimicking theorem for conditioned Ito processes and applies it to confirm a conjecture by P.L. Lions on optimal control of such processes, advancing theoretical understanding.

Contribution

It introduces a mimicking theorem for conditioned processes and uses it to prove a conjecture in optimal control theory, linking stochastic processes and control.

Findings

Established a Markovian projection theorem for conditioned Ito processes

Proved a conjecture of P.L. Lions on optimal control of conditioned processes

Enhanced theoretical framework for stochastic process control

Abstract

We first prove a mimicking theorem (also known as a Markovian projection theorem) for the marginal distributions of an Ito process conditioned to not have exited a given domain. We then apply this new result to the proof of a conjecture of P.L. Lions for the optimal control of conditioned processes.