Stochastic Processes under Parameter Uncertainty

TL;DR

This paper develops a framework for nonlinear stochastic processes with uncertain parameters, extending classical martingale problems and introducing new classes of stochastic PDEs, with applications to nonlinear Lévy processes.

Contribution

It introduces a general framework for nonlinear expectations under parameter uncertainty, extending martingale problem methods and modeling nonlinear Lévy processes and stochastic PDEs.

Findings

Proves the dynamic programming principle for the nonlinear expectations.

Establishes conditions for the USC_b-Feller property.

Develops a Markov selection principle under parameter uncertainty.

Abstract

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a stochastic process with uncertain parameters. We develop a general framework which can be seen as a version of the martingale problem method of Stroock and Varadhan with parameter uncertainty. To illustrate our methodology, we explain how it can be used to model nonlinear L\'evy processes in the sense of Neufeld and Nutz, and we introduce the new class of stochastic partial differential equations under parameter uncertainty. Moreover, we study properties of the nonlinear expectations. We prove the dynamic programming principle, i.e., the tower property, and we establish conditions for the (strong) $\textit{USC}_b$-Feller property and a strong Markov selection principle.