Nonlinear Semimartingales and Markov Processes with Jumps

TL;DR

This paper investigates nonlinear semimartingales with jumps, establishing their control problems, regularity, and connection to nonlinear PDEs, with applications to a G-Lévy process setting.

Contribution

It introduces a framework for nonlinear semimartingales with uncertain local characteristics and links control problems to viscosity solutions of nonlinear PDEs.

Findings

Control problem coincides with weak and relaxed versions.

Regularity properties of the value function are established.

Conditions for the semigroup to be a viscosity solution are provided.

Abstract

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a semimartingale with uncertain local characteristics. Here, the differential characteristics 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 nonlinear 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 $G$-double exponential L\'evy setting.