scholarly journals Crandall–Lions viscosity solutions for path-dependent PDEs: The case of heat equation

Bernoulli ◽  
2022 ◽  
Vol 28 (1) ◽  
Author(s):  
Andrea Cosso ◽  
Francesco Russo
Keyword(s):  
Heat Equation ◽  
Viscosity Solutions ◽  
Path Dependent

scholarly journals Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs

2020 ◽  
Vol 58 (1) ◽  
pp. 277-302
Author(s):  
Zhenjie Ren ◽  
Nizar Touzi ◽  
Jianfeng Zhang
Keyword(s):  
Viscosity Solutions ◽  
Path Dependent

scholarly journals Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I

2016 ◽  
Vol 44 (2) ◽  
pp. 1212-1253 ◽  
Author(s):  
Ibrahim Ekren ◽  
Nizar Touzi ◽  
Jianfeng Zhang
Keyword(s):  
Viscosity Solutions ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic ◽  
Path Dependent

scholarly journals On viscosity solutions of path dependent PDEs

2014 ◽  
Vol 42 (1) ◽  
pp. 204-236 ◽  
Author(s):  
Ibrahim Ekren ◽  
Christian Keller ◽  
Nizar Touzi ◽  
Jianfeng Zhang
Keyword(s):  
Viscosity Solutions ◽  
Path Dependent

scholarly journals Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Sixian Jin ◽  
Henry Schellhorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Malliavin Calculus ◽  
Series Representation ◽  
Parabolic Type ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Path Dependent

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

scholarly journals Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II

2016 ◽  
Vol 44 (4) ◽  
pp. 2507-2553 ◽  
Author(s):  
Ibrahim Ekren ◽  
Nizar Touzi ◽  
Jianfeng Zhang
Keyword(s):  
Viscosity Solutions ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic ◽  
Path Dependent

scholarly journals Viscosity Solutions of Path-Dependent PDEs with Randomized Time

2020 ◽  
Vol 52 (2) ◽  
pp. 1943-1979
Author(s):  
Zhenjie Ren ◽  
Mauro Rosestolato
Keyword(s):  
Viscosity Solutions ◽  
Path Dependent

scholarly journals Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Rainer Buckdahn ◽  
Christian Keller ◽  
Jin Ma ◽  
Jianfeng Zhang
Keyword(s):  
Viscosity Solutions ◽  
Comparison Principle ◽  
General Setting ◽  
Classical Solutions ◽  
Local Theory ◽  
Stochastic Pdes ◽  
Unified Framework ◽  
Fully Nonlinear ◽  
Smooth Test ◽  
Path Dependent

Abstract We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.

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