New Kinds of High-Order Multistep Schemes for Coupled Forward Backward Stochastic Differential Equations

2014 ◽  
Vol 36 (4) ◽  
pp. A1731-A1751 ◽  
Author(s):  
Weidong Zhao ◽  
Yu Fu ◽  
Tao Zhou
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
High Order

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.

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.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

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