Published in International Journal of Stochastic Analysis, Vol. 2014 at doi: 10.1155/2014/852962


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, β„Žβˆ—, we specify variable (adaptive) step sizes relative to β„Žβˆ— 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 𝑂(β„Žβˆ—). An order of error 𝑂(β„Žβˆ—π‘) for any 𝑝 < 1 is obtained when the approximation is run up to a time within β„Žβˆ—π‘ž of the singularity for an appropriate choice of exponent π‘ž. 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 𝑂(β„Žβˆ—), independent of how close to the singularity the approximation is considered.