Abstract
This article shows that the solution of a backward stochastic differential equation under G-expectation provides a probabilistic interpretation for the viscosity solution of a type of path-dependent Hamilton-Jacobi-Bellman equation. Particularly, a G-martingale can be considered as a nonlinear path-dependent partial differential equation (PDE). We also show that certain class of path-dependent PDEs can be transformed into classical multiple state-dependent PDEs. As an application, the path-dependent uncertain volatility model can be described directly by path-dependent Black-Scholes-Barrenblett equations.
AMS Subject Classification:
Acknowledgments
The author would like to offer thanks to S. Peng for his helpful suggestions and to the reviewer for a careful reading and insightful comments. This work is partially supported by the Project 111 (No. B12023), NSF of China (No. 10921101) and the National Basic Research Program of China (973 Program) (No. 2007CB814900) (Financial Risk).
Notes
Right continuous and left limited.