Nonlinear Continuous Semimartingales

David Criens; Lars Niemann

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arXiv · 2204.07823

Nonlinear Continuous Semimartingales

David Criens·Lars Niemann

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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.

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David Criens, Lars Niemann. 2022-04-16. Nonlinear Continuous Semimartingales. https://arxiv.org/abs/2204.07823

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