Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 3

Stochastic Models with Brownian Motion and White Noise

 

 

In the previous chapter, we have learnt that a Brownian motion can be used as:

1) an approximation of a discrete-time process (for example, a process of sums of independent random variables) as space and time scales infinitely increase with an appropriate ratio;

2) a mathematical idealization of (the integral of) a high-intensity and short-memory continuous-time random process (noise).

These two cases of applications of Brownian motion can be essentially extended.

3.1. Discrete time

Discrete-time (non-random) sequences , ∈ ₊, describing, for example, biological, physical, chemical or financial processes are often defined by recurrent equations:

[3.1]

Consider, for example, the following simplest model describing the dynamics of the population growth:

[3.2]

where λ is the coefficient showing the reproduction intensity of population. In this model, we do not take into account the fact that a population cannot grow infinitely, and its reproduction intensity decreases for various reasons (limited food resources, diseases, ...