Monte Carlo-type simulation for solving stochastic ordinary differential equations

We outline a method for solving numerically initial-value and boundary-value problems for ordinary differential equations whose coefficients and/or initial and boundary data are random quantities. The method consists of simulating on the computer several realizations of the stochastic processes that appear in the coefficients of the equations and similarly for the data. Such a simulation is based upon generating suitable sequences of random numbers, for which reason the method can be thought of as a Monte Carlo method. We then solve “pathwise” the equation and finally compute the quantities of interest such as expected values, moments, etc. over such realizations. The numerical error can also be estimated. The method is very simple but quite general, moreover it respects the correct probabilistic structure of the problem and is trivially suited to parallel implementation.