Time-symmetric optimal stochastic control problems in space-time domains

TL;DR

This paper introduces a novel class of time-symmetric stochastic control problems on space-time domains, utilizing dual adjoint stopping times and free-boundary PDEs, extending Schrödinger's problem to new settings.

Contribution

It develops a new framework for time-symmetric stochastic processes using dual adjoint stopping times and free-boundary PDEs, generalizing Schrödinger's problem to space-time domains.

Findings

Characterization of stochastic processes via dual adjoint stopping times

Derivation of associated free-boundary PDEs

Connection to hidden diffusion processes

Abstract

We present a pair of adjoint optimal control problems characterizing a class of time-symmetric stochastic processes defined on random time intervals. The associated PDEs are of free-boundary type. The particularity of our approach is that it involves two adjoint optimal stopping times adapted to a pair of filtrations, the traditional increasing one and another, decreasing. They are the keys of the time symmetry of the construction, which can be regarded as a generalization of "Schr\"odinger's problem" (1931-32) to space-time domains. The relation with the notion of "Hidden diffusions" is also described.