An Explicit Second-Order Numerical Scheme to Solve Decoupled Forward Backward Stochastic Equations

2014 ◽  
Vol 4 (4) ◽  
pp. 368-385 ◽  
Author(s):  
Yu Fu ◽  
Weidong Zhao
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
High Accuracy ◽  
Stochastic Equations ◽  
Explicit Scheme ◽  
Second Order ◽  
General Error

AbstractAn explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

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 A New Simplified Weak Second-Order Scheme for Solving Stochastic Differential Equations with Jumps

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 224
Author(s):  
Yang Li ◽  
Yaolei Wang ◽  
Taitao Feng ◽  
Yifei Xin
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
Second Order ◽  
Trapezoidal Rule ◽  
Order Convergence ◽  
Integration By Parts Formula ◽  
Order Scheme ◽  
Second Order Convergence

In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, we theoretically prove that the numerical scheme has second-order convergence rate. To demonstrate the effectiveness and the second-order convergence rate, three numerical experiments are given.

Prediction-Correction Scheme for Decoupled Forward Backward Stochastic Differential Equations with Jumps

2016 ◽  
Vol 6 (3) ◽  
pp. 253-277 ◽  
Author(s):  
Yu Fu ◽  
Jie Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Measure ◽  
Backward Stochastic Differential Equations ◽  
Explicit Scheme ◽  
Poisson Random Measure ◽  
Correction Scheme ◽  
Compensated Poisson Random Measure ◽  
Theoretical Results ◽  
General Error

AbstractBy introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.

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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