Numerical solutions to stochastic differential equations with constant diffusion coefficients

Numerical solutions to stochastic differential equations are not easy to develop because of the nondifferentiability of the random motion involved. In this work, the special properties of Brownian motion (the random process) are presented, along with the problems they create for both the numerical solutions and the evaluation of the errors of the solutions to the equations of which they are a part. Runge-Kutta methods are developed because they do not involve high-order derivatives of the differential equation, and their order of convergence is shown.

Finally, some numerical examples are presented, along with the results from two Runge-Kutta methods. The methods involve the use of Monte-Carlo methods for observing the mean and variance of the solutions.