Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 11

Examples

 

 

In this chapter, we give some examples illustrating the theory that has been presented thus far. The motivation of their choice is twofold. First, the equations considered are important and used in the physical sciences and finance. Second, they clearly show that outer perturbations (“noises”) do not only quantitatively change the behavior of a macroscopic system by augmenting its deviations from the average behavior which is usually described by deterministic equations; we shall see that sufficiently strong multiplicative noises can induce qualitative changes in the behavior of the system that in principle cannot be described by ordinary (non-stochastic) equations that are oriented toward the average behavior.

In the figures, we present several typical trajectories of various stochastic differential equations. In the next chapter we shall see how such trajectories are obtained by a computer simulation. However, here we want to emphasize that though those trajectories are typical, they are random, since they depend on the random trajectories of a Brownian motion. Therefore, each time we run a computer program that solves or simulates an SDE, on a display (or on a paper sheet), we see another trajectory. However, the trajectories of a Brownian motion are usually simulated by means of sequences of the so-called pseudorandom numbers, that is, sequences of non-random numbers having properties of sequences of random numbers. Usually, a computer program allows ...