Rounding Error in Numerical Solution of Stochastic Differential Equations

The present investigation is concerned with estimating the rounding error in numerical solution of stochastic differential equations. A statistical rounding error analysis of Euler's method for stochastic differential equations is performed. In particular, numerical evaluation of the quantities E|X(t n )−Yˆ n |2 and E[F(Yˆ n )−F(X(t n ))] is investigated, where X(t n ) is the exact solution at the nth time step and Yˆ n is the approximate solution that includes computer rounding error. It is shown that rounding error is inversely proportional to the square root of the step size. An extrapolation technique provides second-order accuracy, and is one way to increase accuracy while avoiding rounding error. Several computational results are given.