Nonlinear Continuous Semimartingales
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.
Cite
@article{arxiv.2204.07823,
title = {Nonlinear Continuous Semimartingales},
author = {David Criens and Lars Niemann},
journal= {arXiv preprint arXiv:2204.07823},
year = {2023}
}