Brownian motion is a fundamentally important stochastic process in that it is a central notion throughout the theoretical development of stochastic processes. In this chapter, we present the definition and some properties of Brownian motion.

**Definition 28.1** (Standard Brownian Motion). Let (Ω, , *P*) be a probability space. A stochastic process {*B*_{t} : *t* ≥ 0} on (Ω, , *P*) is called a *standard Brownian motion* if it satisfies the following conditions:

(a) *P*{*B*_{0} = 0} = *P*{ω : *B*_{0}(ω) = 0} = 1.

(b) For any 0 ≤ *s* < *t*, the increment *B*_{t} − *B*_{s} is a random variable normally distributed with mean 0 and variance *t* − *s*, that is, for any *a* < *b*, we have

(c) *B*_{t} has independent increments, that is, for any 0 ≤ *t*_{1} < *t*_{2} < … < *t*_{n}, the random variables *B*_{t1}, *B*_{t2} − *B*_{t1},…, *B*_{tn} − *B*_{tn−1} are independent.

(d) Almost all sample paths of *B*_{t} are continuous functions:

**Definition 28.2** (Brownian Motion with Respect to Filtrations). Let {_{t} : *t* ≥ 0} be a filtration. A stochastic ...

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