Deferred Correction Methods for Forward Backward Stochastic Differential Equations

2017 ◽  
Vol 10 (2) ◽  
pp. 222-242 ◽  
Author(s):  
Tao Tang ◽  
Weidong Zhao ◽  
Tao Zhou
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
Classical Method ◽  
Practical Importance ◽  
Backward Stochastic Differential Equations ◽  
Euler Method ◽  
High Order ◽  
Deferred Correction ◽  
Key Features

AbstractThe deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.

A Multistep Scheme for Decoupled Forward-Backward Stochastic Differential Equations

2016 ◽  
Vol 9 (2) ◽  
pp. 262-288 ◽  
Author(s):  
Weidong Zhao ◽  
Wei Zhang ◽  
Lili Ju
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Error Estimate ◽  
Stochastic Differential Equations ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Backward Stochastic Differential Equations ◽  
Multistep Scheme ◽  
Theoretical Results ◽  
General Error

AbstractUpon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

scholarly journals High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Quan Zhou ◽  
Yabing Sun
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Taylor Expansion ◽  
Backward Stochastic Differential Equations ◽  
High Order ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
One Step

<p style='text-indent:20px;'>In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.</p>

scholarly journals High-order Combined Multi-step Scheme for Solving Forward Backward Stochastic Differential Equations

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Long Teng ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Finite Difference Methods ◽  
Gaussian Quadrature ◽  
Backward Stochastic Differential Equations ◽  
High Order ◽  
Step Method ◽  
Numerical Illustration ◽  
Combination Technique ◽  
Conditional Expectations

AbstractIn this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we propose a new multi-step scheme by adopting the high-order multi-step method in Zhao et al. (SIAM J. Sci. Comput., 36(4): A1731-A1751, 2014) with the combination technique. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by using the finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the temporal direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.

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