Upper envelopes of families of Feller semigroups and viscosity solutions to a class of nonlinear Cauchy problems

Nendel, Max; Röckner, Michael

doi:10.48550/arxiv.1906.04430

Cited by 4 publications

(7 citation statements)

References 13 publications

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“…The main tools, we use in our analysis, are convex duality, so-called Nisio semigroups (cf. Nisio [21], Denk et al [10], Nendel and Röckner [19]) and a convex version of Kolmogorov's extension theorem, see Denk et al [11]. Restricting the time parameter in the present work to the set of natural numbers leads to a discrete-time Markov chain, in the sense of [11,Example 5.3].…”

Section: Introduction and Main Resultsmentioning

confidence: 91%

Markov chains under nonlinear expectation

Nendel¹

2018

Preprint

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In this paper, we consider continuous-time Markov chains with a finite state space under nonlinear expectations. We define so-called Q-operators as an extension of Q-matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q-operators in terms of a positive maximum principle, a dual representation by means of Q-matrices, continuous-time Markov chains under convex expectations and nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive a primal and dual representation of the convex semigroup arising from a Markov chain under a convex expectation via the Fenchel-Legendre transformation of its generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

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Section: Introduction and Main Resultsmentioning

confidence: 91%

Markov chains under nonlinear expectation

Nendel¹

2018

Preprint

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“…Using a standard procedure from Denk et al [14], Röckner and Nendel [26], or Bartl et al [6], which can be almost literally adapted to our setup, one can show that the map t → S(t)u 0 , for fixed u 0 ∈ C b , is a viscosity solution to (4.6) using the following notion of a viscosity solution.…”

Section: Relaxation Of Assumption (A1)mentioning

confidence: 99%

“…The present construction is related to the one of the Nisio semigroup [28], see Section 4.6, where the family (I(t)) t≥0 is replaced by a supremum over a class of linear semigroups, cf. Denk et al [14] and Nendel and Röckner [26]. In contrast to the Nisio semigroup, where I(π) is increasing over refining partitions, the Wasserstein robust semigroup is obtained as a decreasing limit.…”

Section: Introductionmentioning

confidence: 99%

Wasserstein perturbations of Markovian transition semigroups

Fuhrmann¹,

Kupper²,

Nendel³

2021

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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.

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“…In fact, by a Banach space valued version of Jensen's inequality (see e.g. [14] or [27]) and the translation invariance of S,…”

Section: Invariant Domainsmentioning

confidence: 99%

“…for a suitable family (A λ ) λ∈Λ of generators and a nonempty control set Λ, have been studied using a semigroup-theoretic approach, cf. Denk et al [14] and Nendel and Röckner [27]. We would like to point out, choosing A λ := λ 2 2 ∂ uu for λ ∈ Λ := [σ, σ], the G-heat equation (1.1) is of the form (1.2).…”

Section: Introductionmentioning

confidence: 99%

Convex monotone semigroups on lattices of continuous functions

Denk,

Kupper,

Nendel

2020

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We consider convex monotone semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a σ-Dedekind complete Banach lattice with an additional assumption on the dual space. As typical examples, we consider the space of bounded uniformly continuous functions and the space of continuous functions vanishing at infinity. We show that the domain of the classical generator for convex monotone C0-semigroups, which is defined in terms of the time derivative at 0 w.r.t. the supremum norm, is typically not invariant. We thus propose alternative forms of generators and domains, for which we prove the invariance under the semigroup. As a consequence, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are discussed in several examples related to fully nonlinear partial differential equations, such as uncertain shift semigroups and semigroups related to G-heat equations (fully nonlinear versions of the heat equation).

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Upper envelopes of families of Feller semigroups and viscosity solutions to a class of nonlinear Cauchy problems

Cited by 4 publications

References 13 publications

Markov chains under nonlinear expectation

Markov chains under nonlinear expectation

Wasserstein perturbations of Markovian transition semigroups

Convex monotone semigroups on lattices of continuous functions

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