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.
Funding Statement
DC acknowledges financial support from the DFG project No. SCHM 2160/15-1. LN acknowledges financial support from the DFG project SCHM 2160/13-1.
Acknowledgments
We thank the anonymous referee for many helpful comments and suggestions. Moreover, we are grateful to Andrea Cosso for many helpful comments related to the preprint [4].
Citation
David Criens. Lars Niemann. "Nonlinear continuous semimartingales." Electron. J. Probab. 28 1 - 40, 2023. https://doi.org/10.1214/23-EJP1037
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