#### 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, β^{β}, we specify variable (adaptive) step sizes relative to β^{β} 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 π(β^{β}). An order of error π(β^{βπ}) for any π < 1 is obtained when the approximation is run up to a time within β^{βπ} of the singularity for an appropriate choice of exponent π. 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 π(β^{β}), independent of how close to the singularity the approximation is considered.

#### Recommended Citation

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).

## Comments

Published in

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