An overview of Viscosity Solutions of Path-Dependent PDEs
Abstract
This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].
- Publication:
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arXiv e-prints
- Pub Date:
- August 2014
- DOI:
- 10.48550/arXiv.1408.5267
- arXiv:
- arXiv:1408.5267
- Bibcode:
- 2014arXiv1408.5267R
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Probability
- E-Print:
- Stochastic Analysis and Applications, 2014, 100 (2014), 397-453