Note on Viscosity Solution of Path-Dependent PDE and G-Martingales
By Shige Peng. 2011
68 citations recorded by the source corpus.
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.
Subfields: Finance, Management Science and Operations Research, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/d24112b14dd0d6bacfd9b8c0daf8e06459c8b330
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In Finance
- Falei Wang
- Shige Peng
- Jianfeng Zhang
- Li-Xin Zhang
- Marcel Nutz
- Yongsheng Song
- Mingshang Hu
- Nizar Touzi
In Management Science and Operations Research
- Falei Wang
- Shige Peng
- Li-Xin Zhang
- Marcel Nutz
- Mingshang Hu
- Yongsheng Song
- Jianfeng Zhang
- Nizar Touzi
In Numerical Analysis
Citations in the corpus
- Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, unknown): “We remark that Peng [27] also investigated the comparison principle for fully nonlinear PPDEs by using a different approach.”
- BSDE, path-dependent PDE and nonlinear Feynman-Kac formula (2011, background): “Pardoux and Peng [18, 1990] established the existence and uniqueness theorem for nonlinear BSDEs under a standard Lipschitz condition.”
- 7 – Backward Stochastic Differential Equations (2011, unknown): “Besides those and among many others, we note that Lepeltier & San Martin [135] studied BSDEs with non-Lipschitz continuous generators, Tang & Li [223] studied BSDEs driven by jump processes, Fuhrman & Tessitore [95] studied BSDEs in infinite dimensional spaces, Darling & Pardoux [54] studied BSDEs w”