# Numerical Integration of Stochastic Differential Equations

@inproceedings{Seydel2004NumericalIO, title={Numerical Integration of Stochastic Differential Equations}, author={R. Seydel}, year={2004} }

This chapter provides an introduction into the numerical integration of stochastic differential equations (SDEs). Again X t denotes a stochastic process and solution of an SDE,
$$\frac{{\partial y}}{{\partial \tau }} = \frac{{{\partial ^2}y}}{{\partial {x^2}}}$$

#### 142 Citations

Discrete approximation of stochastic differential equations

- Mathematics
- 2010

It is shown how stochastic Itô-Taylor schemes for stochastic ordinary differential equations can be embedded into standard concepts of consistency, stability and convergence. An appropriate choice of… Expand

Numerical solutions of stochastic functional differential equations

- Mathematics
- 2003

In this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the… Expand

Numerical method for stationary distribution of stochastic differential equations with Markovian switching

- Mathematics
- 2005

In principle, once the existence of the stationary distribution of a stochastic differential equation with Markovian switching is assured, we may compute it by solving the associated system of the… Expand

Weak order for the discretization of the stochastic heat equation

- Mathematics, Computer Science
- Math. Comput.
- 2009

The discretization is achieved thanks to finite element methods in space and implicit Euler schemes in time and the rate of convergence is twice the one for pathwise approximations. Expand

Solving parabolic stochastic partial differential equations via averaging over characteristics

- Computer Science, Mathematics
- Math. Comput.
- 2009

The method of characteristics and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used and their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. Expand

Uniformly accurate schemes for oscillatory stochastic differential equations

- Computer Science, Mathematics
- ArXiv
- 2021

Abstract. In this work, we adapt the micro-macro methodology to stochastic differential equations for the purpose of numerically solving oscillatory evolution equations. The models we consider are… Expand

The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

- Mathematics
- 2012

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay… Expand

Functional Integral Approach to the Solution of a System of Stochastic Differential Equations

- Mathematics
- 2018

A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability… Expand

Weak approximation of stochastic partial differential equations: the nonlinear case

- Mathematics, Computer Science
- Math. Comput.
- 2011

It is proved that as it is often the case, the weak order of convergence is twice the strong order and Malliavin calculus is a key ingredient in this proof. Expand

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

- Mathematics
- 2007

Abstract In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop… Expand