Wasserstein perturbations of Markovian transition semigroups

TL;DR

This paper studies how nonparametric Wasserstein perturbations affect Markov processes, leading to nonlinear PDEs and providing bounds on sensitivities, with applications to various stochastic models.

Contribution

It introduces a framework for analyzing Wasserstein perturbations of Markov semigroups, connecting nonparametric uncertainty to nonlinear PDEs and sensitivity analysis.

Findings

Nonparametric Wasserstein perturbations lead to nonlinear PDEs for Markov processes.

The nonlinear transition operators coincide with parametric drift uncertainty in standard cases.

Sensitivity bounds are established for the perturbed semigroups.

Abstract

In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of L\'evy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.