Limits of random walks with distributionally robust transition probabilities

TL;DR

This paper studies a nonlinear random walk with transition probabilities constrained within a Wasserstein neighborhood of a Lévy process, deriving a continuous limit that results in a nonlinear PDE as a perturbation of the Lévy generator.

Contribution

It introduces a novel framework for nonlinear random walks with distributionally robust transitions and explicitly characterizes the resulting PDE in the continuous limit.

Findings

Derived the generator of the nonlinear semigroup

Explicitly computed the PDE as a perturbation of the Lévy generator

Established the connection between discrete nonlinear walks and continuous nonlinear PDEs

Abstract

We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed L\'evy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial L\'evy process.