Upper Envelopes of Families of Feller Semigroups and Viscosity Solutions to a Class of Nonlinear Cauchy Problems
Year: 2021
Type: article
Abstract: In this paper, we consider the (upper) semigroup envelope, i.e., the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract HJB-type PDEs related to stochastic optimal control problems arising in the field of robust finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. The procedure is then applied to numerous examples, in particular, nonlinear PDEs that arise from control problems for infinite-dimensional Ornstein--Uhlenbeck and Lévy processes.
DOI: 10.1137/20m1314823
Authors: Max Nendel, Michael Röckner
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Award: G1428129756
OpenAlex ID: W2952531667