Runge-Kutta and recursive distribution numerical methods for approximate solution of stochastic differential equations
Abukhaled, Marwan Ibrahim
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This research concerns numerical solution of stochastic differential equations and is divided into two different and independent approaches. In the first approach, a class of Runge-Kutta methods is developed, analyzed and numerically tested. It is shown that these methods are of second-order accuracy in the weak sense for estimating expectations of functions of the solution for scalar as well as for systems of stochastic differential equations. It is also shown that in these methods, variance reduction techniques can be applied to reduce the stochastic error involved in estimating the expectations of functions of the solution. These second-order explicit methods are unique in the sense that they do not involve derivatives of the drift and diffusion coefficients and they can be easily programmed and implemented. In the second approach, it is shown that probability distributions of approximate sample paths of the solution satisfy a recursive integral equation. These probability distributions can then be approximated by numerically solving the integral equation. The advantage of this approach is the avoidance of computing thousands of sample paths as is generally the case in most standard numerical methods. This approach is shown to be useful for numerical solution of first-passage time problems.