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International Journal of Stochastic Analysis
Volume 2014 (2014), Article ID 852962, 6 pages
Research Article

Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity

Department of Mathematics, Southern Illinois University Carbondale, 1245 Lincoln Drive, Carbondale, IL 62901, USA

Received 20 December 2013; Revised 27 May 2014; Accepted 10 June 2014; Published 2 July 2014

Academic Editor: M. Jesus Lopez-Herrero

Copyright © 2014 Harry Randolph Hughes and Pathiranage Lochana Siriwardena. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, , we specify variable (adaptive) step sizes relative to which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order . An order of error for any is obtained when the approximation is run up to a time within of the singularity for an appropriate choice of exponent . We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order , independent of how close to the singularity the approximation is considered.