Wasserstein perturbations of Markovian transition 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 Lévy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.

Convex monotone semigroups and their generators with respect to $Γ$-convergence

“…As seen in the proof of [23,Lemma 3.6], it holds lim h↓0 |R(h)g − g|(x) = 0 for all g ∈ Lip b . Hence, we can estimate…”

“…However, condition (vi), which implies continuity from above, also guarantees the semigroup property. Another possibility to ensure the semigroup property is to assume norm convergence or monotone convergence, see [8,23,34]. Remark 4.2.…”

“…Let (µ t ) t≥0 ⊂ P p and (ψ t ) t≥0 be a family of functions ψ t : R d → R d . Following the setting in [23], for every…”

“…Typical examples include Koopman semigroups and transition semigroups of Lévy and Ornstein-Uhlenbeck processes, see [23]. Denote by L R the Lipschitz set of R. Let C ∞ 0 be the space of all infinitely differentiable functions f ∈ C 0 such that all derivatives are in C 0 , where C 0 consists of all continuous functions f : R d → R with lim |x|→∞ f (x) = 0.…”

Fully-dynamic risk measures: horizon risk, time-consistency, and relations with BSDEs and BSVIEs

“…We propose h-longevity to quantify this penalization. The impact of the length of the time horizon on risk evaluation has already been detected in the work [15] on epistemic uncertainty. There, it comes out that ambiguity on the model choice is growing with the time horizon.…”