Nonlinear Continuous Semimartingales

TL;DR

This paper introduces a framework for nonlinear continuous semimartingales with uncertain local characteristics, linking nonlinear expectations to path-dependent PDEs and establishing their unique viscosity solutions.

Contribution

It develops a dynamic programming principle for nonlinear expectations and characterizes the associated value function as a unique viscosity solution of a nonlinear path-dependent PDE.

Findings

Established a dynamic programming principle for nonlinear expectations

Linked the value function to a nonlinear path-dependent PDE

Proved the nonlinear expectation solves a nonlinear martingale problem

Abstract

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued function that depends on time and path in a non-Markovian way. We provide a dynamic programming principle for the nonlinear expectation and we link the corresponding value function to a variational form of a nonlinear path-dependent partial differential equation. In particular, we establish conditions that allow us to identify the value function as the unique viscosity solution. Furthermore, we prove that the nonlinear expectation solves a nonlinear martingale problem, which confirms our interpretation as a nonlinear semimartingale.