Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity. (2nd July 2014) - Hughes, Harry Randolph; Siriwardena, Pathiranage Lochana
Abstract : 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 toh * 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 orderO h * . An order of errorO h * p for anyp < 1 is obtained when the approximation is run up to a time withinh * q of the singularity for an appropriate choice of exponentq . 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 orderO h *, independent of how close to the singularity the approximation is considered.