scholarly journals Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs

2007 ◽  
Vol 60 (7) ◽  
pp. 1081-1110 ◽  
Author(s):  
Patrick Cheridito ◽  
H. Mete Soner ◽  
Nizar Touzi ◽  
Nicolas Victoir
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
Second Order ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

scholarly journals A probabilistic numerical method for fully nonlinear parabolic PDEs

2011 ◽  
Vol 21 (4) ◽  
pp. 1322-1364 ◽  
Author(s):  
Arash Fahim ◽  
Nizar Touzi ◽  
Xavier Warin
Keyword(s):  
Numerical Method ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations

2014 ◽  
Vol 15 (3) ◽  
pp. 618-646 ◽  
Author(s):  
Weidong Zhao ◽  
Wei Zhang ◽  
Lili Ju
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Error Estimates ◽  
Backward Stochastic Differential Equations ◽  
Second Order ◽  
Numerical Schemes ◽  
Reference Equations ◽  
Taylor Scheme ◽  
Theoretical Results

AbstractIn this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

scholarly journals Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 278-310
Author(s):  
Weinan E ◽  
Jiequn Han ◽  
Arnulf Jentzen
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Numerical Algorithms ◽  
Variational Problems ◽  
Least Square ◽  
Picard Iteration ◽  
Parabolic Pdes ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Very High

Abstract In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

2015 ◽  
Vol 18 (5) ◽  
pp. 1482-1503 ◽  
Author(s):  
Tao Kong ◽  
Weidong Zhao ◽  
Tao Zhou
Keyword(s):  
Backward Stochastic Differential Equation ◽  
High Order ◽  
Cauchy Problems ◽  
Numerical Schemes ◽  
Parabolic Pdes ◽  
Hjb Equations ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Nonlinear Parabolic Pde

AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

scholarly journals On a discrete scheme for time fractional fully nonlinear evolution equations

2020 ◽  
Vol 120 (1-2) ◽  
pp. 151-162 ◽  
Author(s):  
Yoshikazu Giga ◽  
Qing Liu ◽  
Hiroyoshi Mitake
Keyword(s):  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Discrete Scheme ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Convergence Of The Scheme

We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo’s time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.

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