Note on Viscosity Solution of Path-Dependent PDE and G-Martingales
arXiv:1106.1144
Abstract
In the 2nd version of this note we introduce the notion of viscosity solution for a type of fully nonlinear parabolic path-dependent partial differential equations (P-PDE). We then prove the comparison theorem (or maximum principle) of this new type of equation which is the key property of this framework. To overcome the well-known difficulty of non-compactness of the space of paths for the maximization, we have introduced a new approach, called left frozen maximization approach which permits us to obtain the comparison principle for smooth as well as viscosity solutions of path-dependent PDE. A solution of a backward stochastic differential equation and a G-martingale under a G-expectation are typical examples of such type of solutions of P-PDE. The maximum principle for viscosity solutions of classical PDE, called state dependent PDE, is a special case.
References in corpus (4)
Cited by in corpus (19)
- On viscosity solutions of path dependent PDEs
- A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations
- Strong-viscosity Solutions: Semilinear Parabolic PDEs and Path-dependent PDEs
- A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance
- Weak Functional Itô Calculus and Applications
- Viscosity Solutions of Path-Dependent PDEs and Non-Markovian Forward-Backward Stochastic Equations
- An extension of the functional Ito formula under a family of non-dominated measures
- Viscosity Solutions to Second Order Path-Dependent Hamilton-Jacobi-Bellman Equations and Applications
- Ergodicity of Sublinear Markovian Semigroups
- Crandall-Lions Viscosity Solutions for Path-Dependent PDEs: The Case of Heat Equation
- Stochastic differential equations driven by G-Brownian motion and ordinary differential equations
- G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE
- Viscosity Solutions of Stochastic Hamilton-Jacobi-Bellman Equations
- A Chen-Fliess approximation for diffusion functionals
- On monotonicity and order-preservation for multidimensional G-diffusion processes
- Pseudo Markovian Viscosity Solutions of Fully Nonlinear Degenerate PPDEs
- Viscosity Solutions to Path-Dependent HJB Equation and Applications
- Viscosity Solutions to Second Order Path-Dependent Hamilton-Jacobi-Bellman Equations in Hilbert Spaces
- Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations