The initial sections of this chapter are devoted to the definition of Brownian motion (the mathematical object, not the physical phenomenon) and a compilation of its basic properties. The properties in question are quite deep, and readers will be referred elsewhere for proofs. Later sections are devoted to the derivation of further properties and to calculation of several interesting distributions associated with Brownian motion.
Before proceeding, readers are advised to at least look through Appendices A and B, which enunciate some standing assumptions (in particular, joint measurability and right-continuity of stochastic processes) and explain several important conventions regarding notation and terminology. As noted there, and in the Guide to Notation and Terminology, the value of a stochastic process X at time t may be written either as Xt or as X(t), depending on convenience. The former notation is generally preferred, but the latter is used when necessary to avoid clumsy typography like subscripts on subscripts.
A stochastic process X is said to have independent increments if the random variables X(t0), X(t1) – X(t0),…, X(tn) – X(tn-1) are independent for any n ≥ 1 and 0 ≤ t0 < … < tn < ∞. It is said to have stationary independent increments if moreover the distribution of X(t) – X(s) depends only on t – s. Finally, we write Z ∼ N(μ,σ2) to mean that the random variable Z has the normal distribution with mean μ and variance σ2.