2019/06/11 by Max Nendel, Michael Röckner, Nendel, Max +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · Social Sciences · #47H20 #49L25 #60G20 #Analysis of PDEs (math.AP) #Economic theories and models #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications #math.AP #math.OC #math.PR #msc:47H20 #msc:49L25 #msc:60G20
paper · pdf · doi:10.48550/arxiv.1906.04430
openalex publication_date 2019/06/11 · arxiv created 2020/11/19 · arxiv updated 2020/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Hamilton-Jacobi-Bellman-type partial differential equations 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.