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 𝑂(ℎ∗). An order of error 𝑂(ℎ∗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 𝑂(ℎ∗), independent of how close to the singularity the approximation is considered.
Hughes, Harry R. and Siriwardena, Pathiranage L. "Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity." (Jul 2014).