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IMA Journal of Numerical Analysis Advance Access originally published online on December 6, 2006
IMA Journal of Numerical Analysis 2007 27(3):479-506; doi:10.1093/imanum/drl032
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An adaptive Euler–Maruyama scheme for SDEs: convergence and stability

H. Lamba{dagger}

Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

J. C. Mattingly{ddagger}

Department of Mathematics, Duke University, Durham, NC 27708, USA

A. M. Stuart§

Mathematics Institute, Warwick University, Coventry, CV4 7AL, UK

{dagger} Email: hlamba{at}gmu.edu

{ddagger} Email: jonm{at}math.duke.edu

§ Corresponding author. Email: stuart{at}maths.warwick.ac.uk

Received on 20 December 2005. Revised on 15 July 2006.


  Abstract

The understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler–Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself.

Key Words: stochastic differential equations; adaptive time discretization; convergence; stability; ergodicity; exponential moment bounds


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