Nonlinear semimartingales and Markov processes with jumpsNonlinear semimartingales and Markov processes with jumpsD. Criens and L. Niemann

In this paper, we study a family of nonlinear (conditional) expectations that can be understood as a semimartingale with uncertain local characteristics, which are prescribed by a time and path-dependent set-valued function. We show that the associated control problem coincides with both its weak and relaxed counterparts. Furthermore, we establish regularity properties of the value function and discuss their relation to Feller properties of sublinear semigroups. In the Markovian case, we provide conditions that allow us to identify the corresponding semigroup as the unique viscosity solution to a nonlinear Hamilton–Jacobi–Bellman equation. To illustrate our results, we discuss a random G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}-double exponential Lévy setting.