scholarly journals A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method

2009 ◽  
Vol 13 (3) ◽  
pp. 415-443 ◽  
Author(s):  
Mariko Ninomiya ◽  
Syoiti Ninomiya
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Approximation Scheme ◽  
Higher Order ◽  
Weak Approximation ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

scholarly journals Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion

2019 ◽  
Vol 17 (1) ◽  
pp. 1515-1525
Author(s):  
Yazid Alhojilan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Transport Theory ◽  
Numerical Solutions ◽  
Optimal Transport ◽  
Stochastic Integrals ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

Implicit Order 3/2 Strong Method for Numerical Solutions of Stratonovich Stochastic Differential Equations

2019 ◽  
Vol 16 (8) ◽  
pp. 3137-3140
Author(s):  
Yazid Alhojilan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Approximation Method ◽  
Taylor Expansion ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Higher Order ◽  
Implicit Method ◽  
Kutta Method ◽  
Runge Kutta Method

Due to that the explicit methods in solving stochastic differential equations give instability and inaccurate results, the aim of this paper is to derive an effective implicit method gives higher-order approximate solutions for a stiff stochastic differential equations by using Runge-Kutta method. It relies on the Stratonovich-Taylor expansion and uses the notion of perturbation and coupling to carry out the method. The validity of this new approximation method is shown by implementing in MATLAB and, showing the convergence of the method graphically.

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