Brownian motion is by far the most important stochastic process. It is the archetype of Gaussian processes, of continuous time martingales, and of Markov processes. It is basic to the study of stochastic differential equations, financial mathematics, and filtering, to name only a few of its applications.

In this chapter we define Brownian motion and consider some of its elementary aspects. Later chapters will take up the construction of Brownian motion and properties of Brownian motion paths.

Definition and basic properties

Let (Ω, ℱ, ℙ) be a probability space and let {ℱt} be a filtration, not necessarily satisfying the usual conditions.

Definition 2.1Wt = Wt(ω) is a one-dimensional Brownian motion with respect to {ℱt} and the probability measure ℙ, started at 0, if

1. (1)Wt is ℱt measurable for each t ≥ 0.

2. (2)W0 = 0, a.s.

3. (3)Wt − Ws is a normal random variable with mean 0 and variance t − s whenever s < t.

4. (4)Wt − Ws is independent of ℱs whenever s < t.

5. (5)Wt has continuous paths.

If instead of (2) we have W0 = x, we say we have a Brownian motion started at x. Definition 2.1(4) is referred to as the independent increments property of Brownian motion. The fact that Wt1 – Ws has the same law as Wt–s, which follows from Definition 2.1(3), is called the stationary increments property. When no filtration is specified, we assume the filtration is the filtration generated by W, i.e., ℱt = σ (Ws; s ≤ t). Sometimes a one-dimensional Brownian motion started at 0 is called a standard Brownian motion.

Often a Markov process is specified in terms of its behavior at each point, and one wants to form a global picture of the process. This means one is given the infinitesimal generator, which is a linear operator that is an unbounded operator in general, and one wants to come up with the semigroup for the Markov process.

We will begin by looking further at semigroups and resolvents, and then define the infinitesimal generator of a semigroup. We will prove the Hille–Yosida theorem, which is the primary tool for constructing semigroups from infinitesimal generators. Then we will look at two important examples: elliptic operators in nondivergence form and Lévy processes.

Semigroup properties

Let S be a locally compact separable metric space. We will take ℬ to be a separable Banach space of real-valued functions on S. For the most part, we will take ℬ to be the continuous functions on S that vanish at infinity (with the supremum norm), although another common example is to let ℬ be the set of functions on S that are in L2 with respect to some measure. We use ∥·∥ for the norm on ℬ.

For the duration of this chapter we will make the following assumption.

Assumption 37.1Suppose that Pt, t ≥ 0, are operators acting on ℬ such that

1. (1) the Ptare contractions: ∥Ptf∥≤∥f∥ for all t ≥ 0 and all f ∈ ℬ,

2. (2) the Ptform a semigroup: PsPt = Pt+sfor all s, t ≥ 0, and

3. (3) the Ptare strongly continuous: if f ∈ ℬ, then Ptf → f as t → 0.

At the opposite extreme from Brownian motion is the Poisson process. This is a process that only changes value by means of jumps, and even then, the jumps are nicely spaced. The Poisson process is the prototype of a pure jump process, and later we will see that it is the building block for an important class of stochastic processes known as Lévy processes.

Definition 5.1 Let {ℱt} be a filtration, not necessarily satisfying the usual conditions. A Poisson process with parameter λ > 0 is a stochastic process X satisfying the following properties:

1. (1) X0 = 0, a.s.

2. (2) The paths of Xt are right continuous with left limits.

3. (3) If s < t, then Xt – Xs is a Poisson random variable with parameter λ(t − s).

4. (4) If s < t, then Xt – Xs is independent of ℱs.

Define Xt− = lims→t,s<tXs, the left-hand limit at time t, and ΔXt = Xt – Xt–, the size of the jump at time t. We say a function f is increasing if s < t implies f(s) ≤ f(t). We use “strictly increasing” when s < t implies f(s) < f(t). We have the following proposition.

Proposition 5.2Let X be a Poisson process. With probability one, the paths of Xt are increasing and are constant except for jumps of size 1. There are only finitely many jumps in each finite time interval.

Suppose S is a metric space.We use Sℕ for the product space S×S×… furnished with the product topology. We may view Sℕ as the set of sequences (x1, x2, …) of elements of S. We use the σ-field on Sℕ generated by the cylindrical sets. Given an element x = (x1, x2, …) of Sℕ, we define πn(x) = (x1, …, xn) ∈ Sn.

We suppose we have a Radon probability measure μn defined on Sn for each n. (Being a Radon measure means that we can approximate μn(A) from below by compact sets; see

Folland (1999) for details.) The μn are consistent if μn+1(A × S) = μn(A) whenever A is a Borel subset of Sn. The Kolmogorov extension theorem is the following.

Theorem D.1Suppose for each n we have a probability measure μnon Sn. Suppose the μn's are consistent. Then there exists a probability measure μ Sℕ such that μ(A × Sℕ) = μn(A) for all A ⊂ Sn.

Proof Define μ on cylindrical sets by μ(A × Sℕ) = μn(A) if A ⊂ Sn. By the consistency assumption, μ is well defined. By the Carathéodory extension theorem, we can extend μ to the σ-field generated by the cylindrical sets provided we show that whenever An are cylindrical sets decreasing to ∅, then μ(An) → 0.

Suppose An are cylindrical sets decreasing to ∅ but μ(An) does not tend to 0; by taking a subsequence we may assume without loss of generality that there exists ε > 0 such that μ(An) ≥ ε for all n. We will obtain a contradiction.

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