Abstract: Mean field games may admit multiple equilibria, which naturally raises the question of equilibrium selection. In this talk, we introduce a trembling-hand perfection refinement for stochastic mean field games formulated through relaxed controlled martingale problems. Perturbations are imposed at the level of population controls: they have full support and are required to remain admissible for the state laws they generate. Trembling-hand perfection is then defined by requiring both the perturbed population behavior and the corresponding optimal responses to converge to the same equilibrium. We prove existence under general continuity, growth, and coercivity assumptions, without requiring compact control spaces or bounded coefficients. Finally, a one-dimensional example shows that the refinement is genuinely selective.                                                                    

Abstract: First-passage time (FPT) describes the time at which a stochastic process reaches a specified threshold for the first time. In many applications, however, the relevant threshold evolves over time, motivating the study of first-passage problems with time-dependent boundaries.                                                                                                                                                            

In this talk, I will consider first-passage time problems for two classes of stochastic processes: stochastic differential equations (SDEs) and piecewise-diffusion Markov processes (PDifMPs), with a particular focus on exact simulation. For SDEs with a constant threshold, an exact simulation approach can be constructed using the known first-passage time distribution of Brownian motion as a proposal distribution. A candidate first-passage time is generated from the corresponding Brownian motion and accepted as the first-passage time of the target process according to an acceptance probability derived via Girsanov’s theorem.                                                                                                                                                                

We extend this approach to SDEs with time-dependent thresholds and then use this extension to develop an exact simulation method for PDifMPs with time-dependent thresholds. The resulting framework enables direct simulation of first-passage times without requiring the simulation of entire sample paths over a prescribed time interval and without introducing time-discretisation error.