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, h*, we specify variable (adaptive) step sizes relative to h* 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 Oh*. An order of error Oh*p for any p<1 is obtained when the approximation is run up to a time within h*q of the singularity for an appropriate choice of exponent q. 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 Oh*, independent of how close to the singularity the approximation is considered.
Step size control for the uniform approximation of systems of stochastic differential equations with additive noise
