Limits of random walks with distributionally robust transition probabilities

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 Lévy 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 Lévy process.

Wasserstein perturbations of Markovian transition semigroups

“…In particular, under a proper definition of proximity (e.g., for a geometric Brownian motion, we consider a logarithmic version of the Wasserstein distance), the uncertainty in the generator does not depend on the order of the Wasserstein distance. Our results in Section 3 extend and complement the work [6] as follows. While the analysis in [6] is performed on the space C 0 endowed with the supremum norm, we are working here with locally uniform convergence on a suitable space of (linearly) bounded continuous functions.…”

“…Our results in Section 3 extend and complement the work [6] as follows. While the analysis in [6] is performed on the space C 0 endowed with the supremum norm, we are working here with locally uniform convergence on a suitable space of (linearly) bounded continuous functions. Moreover, our construction allows for state-dependent dynamics such as geometric Brownian motions or Ornstein-Uhlenbeck processes.…”

“…One way to tackle such situations is to consider perturbations of a reference model using a certain notion of proximity, e.g., in terms of Wasserstein distances, cf. Bartl et al [5], Blanchet and Murthy [8], Mohajerin Esfahani and Kuhn [25], Gao and Kleywegt [19], Pflug and Wozobal [31], Zhao and Guan [40], as well as Bartl et al [6] for a dynamic setting.…”

“…The starting point of the present article is the work of Bartl et al [6], where perturbations of certain Lévy processes in R d were considered. Our focus is on more general classes of stochastic processes with values in infinite-dimensional state spaces and their transition semigroups under nonparametric uncertainty.…”

“…Yet, in infinite-dimensional situations, it enables us to treat, for example, Ornstein-Uhlenbeck processes with an unbounded generator in the drift term. A minor yet subtle issue is the a priori chosen dyadic decomposition (partition) of the time interval [0, t] for the definition of the operators (S(t)) t≥0 in [6]. In the present work, we also use a dyadic partitions for the definition of the nonlinear semigroup.…”