## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 12, Number 2 (2002), 664-690.

### The optimal uniform approximation of systems of stochastic differential equations

#### Abstract

We analyze numerical methods for the pathwise approximation of a system of stochastic differential equations. As a measure of performance we consider the $q$th mean of the maximum distance between the solution and its approximation on the whole unit interval. We introduce an adaptive discretization that takes into account the local smoothness of every trajectory of the solution. The resulting adaptive Euler approximation performs asymptotically optimal in the class of all numerical methods that are based on a finite number of observations of the driving Brownian motion.

#### Article information

**Source**

Ann. Appl. Probab., Volume 12, Number 2 (2002), 664-690.

**Dates**

First available in Project Euclid: 17 July 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1026915620

**Digital Object Identifier**

doi:10.1214/aoap/1026915620

**Mathematical Reviews number (MathSciNet)**

MR1910644

**Zentralblatt MATH identifier**

1019.65009

**Subjects**

Primary: 65U05

Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

**Keywords**

Systems of stochastic differential equations pathwise uniform approximation asymptotic optimality adaptive method

#### Citation

Müller-Gronbach, Thomas. The optimal uniform approximation of systems of stochastic differential equations. Ann. Appl. Probab. 12 (2002), no. 2, 664--690. doi:10.1214/aoap/1026915620. https://projecteuclid.org/euclid.aoap/1026915620