ABSTRACT
Stochastic processes continuous in time and in state are presented in this chapter. In the first section, it is shown that the random walk model leads to a diffusion process known as Brownian motion, also referred to as the Wiener process. A diffusion process is defined in Section 8.4, a Markov process having continuous sample paths with the additional property that the infinitesimal mean and variance are finite. It is shown that the transition probability density function of a diffusion process is a solution of the forward and backward Kolmogorov differential equations, the continuous analogues of the Kolmogorov differential equations from Markov chain theory. A more general definition of the Wiener process is given in Section 8.6 that leads to the formulation of Itoˆ stochastic integrals and differential equations, equations whose solutions are stochastic realizations of a diffusion process. Then numerical methods for solution of Itoˆ stochastic differential equations are discussed. The chapter ends with an application of Itoˆ stochastic differential equations to the problem of monitoring the drug concentration in a body.
