Communications in Mathematical Sciences

Volume 9 (2011)

Number 1

A weak trapezoidal method for a class of stochastic differential equations

Pages: 301 – 318

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n1.a15

Authors

David F. Anderson (Department of Mathematics, University of Wisconsin-Madison)

Jonathan C. Mattingly (Duke University, Durham, North Carolina)

Abstract

We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated Itô integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order.

Keywords

weak trapezoidal, error analysis, reaction networks, higher-order methods, stochastic differential equations, numerical methods

2010 Mathematics Subject Classification

60H35, 65C30, 65L20

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