Speaker: Prof. Jian-Guo Liu, Duke University

Date: Dec 8, Friday, 2023, 3:30PM-5:00PM

Venue: 本部览秀楼105学术报告厅

Abstract: Among various rare events, effectively computing transition paths that connect metastable states in a stochastic model remains a crucial problem. In this talk, I will present a stochastic optimal control formulation for transition path problems in an infinite time horizon, specifically for Markov jump processes on Polish spaces. An unbounded terminal cost, applied at a prescribed stopping time, along with a controlled transition rate for the jump process, regulates the transitions between metastable states. Notably, the running cost adopts an entropic form for the control velocity, contrasting with the quadratic form typically used for diffusion processes.

Employing the Girsanov transform, tailored for Markov jump processes, this optimal control problem can be framed within a unified approach. This approach applies to both finite and infinite time horizons with stopping time and involves an optimal change of measures. These measures are defined on the space of càdlàg paths, relative to the law of a reference process. We demonstrate that the committer function, which resolves a backward equation with specific boundary conditions, provides an explicit formula for the optimal path measure. This formula also delineates the associated optimal control for the transition path problem. The unbounded terminal cost results in a singular transition rate, characterized by an unbounded control velocity. We approach this through approximation via Gamma-convergence. The resulting limit path measure effectively solves a Martingale problem, featuring an optimally controlled transition rate. The associated optimal control is linked to the Doob-h transform. Consequently, the optimally controlled process realizes the transition paths with high certainty, achieving this without altering the bridges of the reference process.