Basic Stochastic Computational Methods

In this chapter, several fundamental stochastic computational tools are introduced. The chapter starts by presenting the ideas of the Monte Carlo method, a widely utilized technique that exploits the repeated sampling of random variables to solve many deterministic and stochastic problems. Then the Euler-Maruyama and Milstein schemes are introduced, which are essential methods for numerically finding the path-wise solution of the SDEs. With these tools in hand, the Monte Carlo simulation is combined with the numerical schemes that allow using the ensemble method to approximate the statistics of the SDE numerically. Ergodicity is also discussed. It provides a more efficient way to compute the equilibrium statistics if the underlying SDE has such a desirable feature. Finally, the kernel density estimation is presented, which advances the recovery of a smoothed PDF using only a finite number of samples.