Stochastic Changes

Abstract

The characteristic features of solutions of several types of difference equations were discussed in Chap. 5 to see which sort of processes can be described in this way. These discussions did not account for randomness. However, randomness effects are relevant to many applications. Examples are given by the diffusion of substances in the atmosphere or water, the chaotic motion of molecules and other particles in fluids, and the development of population densities in time. Therefore, the deterministic methods presented in Chap. 5 will be extended in this chapter by the inclusion of randomness effects. The relevance of the stochastic difference equations considered in the following is that these equations provide the solutions of stochastic differential equations and equations for PDFs, which are the basic equations for the modeling of the evolution of stochastic processes. Hence, this chapter provides the basis for the discussions in Chaps. 8 and 10. In particular, the basic ingredient of stochastic evolution equations (the Wiener process) and the basic methodology for the solution of stochastic evolution equations (Monte Carlo simulation) will be introduced in this chapter.