author:
- Peter Kloeden and Andreas Neuenkirch
title: Convergence of Numerical Methods for Stochastic Differential Equations in Mathematical Finance
---

Introduction
============

Consider the Itô stochastic differential equation (SDE) in $\mathbb{R}^{d}$ $$\label{itosde}
dX_t = a(X_t)dt + \sum_{j=1}^{m} b_{j}(X_t)  dW^{(j)}_t,  \quad t \in [0,T], \qquad X_0=x_0 \in \mathbb{R}^d$$ with drift and diffusion coefficients $a$, $b_{j}$ $:$ $\mathbb{R}^{d}$ $\rightarrow$ $\mathbb{R}^{d}$ for $j=1, \ldots, m$. Here $W_t$ $=$ $(W_t^{(1)}, 
\ldots, W_t^{(m)})$, $t \geq 0$, is an $m$-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and superscripts in brackets label components of vectors. Throughout this article it will always be assumed that equation (\[itosde\]) has a unique strong solution.

Explicit solutions of such equations are rarely known, thus one has to rely on numerical methods to simulate their sample paths $X_t(\omega)$ or to estimate functionals $\mathbf{E}\Phi(X)$ for some $\Phi: C([0,T];\mathbb{R}^d) \rightarrow \mathbb{R}$. Typically, such a numerical method relies on a discretization $$0 \leq t_1 \leq t_2 \leq  \ldots \leq t_n =T$$ and a global approximation on $[0,T]$ is obtained by interpolation.

In the case of the classical weak approximation the error of an approximation $\overline{X}$ to $X$ is measured by the quantity $$|\mathbf{E} \phi(X_T) - \mathbf{E} \phi(\overline{X}_T) |$$ for smooth functions $\phi: 