Brownian Motion and Stochastic Calculus

A stochastic process is a mathematical model for the occurrence, at each moment after the initial time, of a random phenomenon. The randomness is captured by the introduction of a measurable space (Ω, ℱ), called the sample space, on which probability measures can be placed. Thus, a stochastic process is a collection of random variables X = {X_t; 0≤t<∞} on (Ω, ℱ), which take values in a second measurable space (S, S ) called the state space. For our purposes, the state space (SS) will be the d-dimensional Euclidean space equipped with the σ-field of Borel sets, i.e., S= ℝ^d, S= ℬ (ℝ^d), where ℬ(U) will always be used to denote the smallest a-field containing all open sets of a topological space U. The index t ∈ [O, ∞) of the random variables X, admits a convenient interpretation as time.

Brownian movement is the name given to the irregular movement of pollen, suspended in water, observed by the botanist Robert Brown in 1828. This random movement, now attributed to the buffeting of the pollen by water molecules, results in a dispersal or diffusion of the pollen in the water. The range of application of Brownian motion as defined here goes far beyond a study of microscopic particles in suspension and includes modeling of stock prices, of thermal noise in electrical circuits, of certain limiting behavior in queueing and inventory systems, and of random perturbations in a variety of other physical, biological, economic, and management systems. Furthermore, integration with respect to Brownian motion, developed in Chapter 3, gives us a unifying representation for a large class of martingales and diffusion processes. Diffusion processes represented this way exhibit a rich connection with the theory of partial differential equations (Chapter 4 and Section 5.7). In particular, to each such process there corresponds a second-order parabolic equation which governs the transition probabilities of the process.

A tremendous range of problems in the natural, social, and biological sciences came under the dominion of the theory of functions of a real variable when Newton and Leibniz invented the calculus. The primary components of this invention were the use of differentiation to describe rates of change, the use of integration to pass to the limit in approximating sums, and the fundamental theorem of calculus, which relates the two concepts and thereby makes the latter amenable to computation. All of this gave rise to the concept of ordinary differential equations, and it is the application of these equations to the modeling of real-world phenomena which reveals much of the power of calculus.

There is a rich interplay between probability theory and analysis, the study of which goes back at least to Kolmogorov (1931). It is not possible in a few sections to develop this subject systematically; we instead confine our attention to a few illustrative cases of this interplay. Recent monographs on this subject are those of Doob (1984) and Durrett (1984).

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

This chapter is an in-depth study of the Brownian local time first encountered in Section 3.6. Our approach to this subject is motivated by the desire to perform computations. This is manifested by the inclusion of the conditional Laplace transform formulas of D. Williams (Subsections 6.3.B, 6.4.C), the derivation of the joint density of Brownian motion, its local time at the origin and its occupation time of the positive half-line (Subsection 6.3.C), and the computation of the transition density for Brownian motion with two-valued drift (Section 6.5). This last computation arises in the problem of controlling the drift of a Brownian motion, within prescribed bounds, so as to keep the controlled process near the origin.