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.

A European call option is the option to buy a share of stock at a given price at some particular time in the future. For example, I might buy a call option to purchase one share of Company X for $40 three months from today. When the three months is up, I check the price of Company X. If, say, it is $35, then my option is worthless, because why would I buy a share for $40 using the option when I could buy it on the open market for $35? But if three months from now, the share price is, say, $45, then I can exercise my option, which means I buy a share for $40, and I can then turn around immediately and sell that share for $45 and make a profit of $5. Thus, today, there is a potential for a profit if I have a call option, and so I should pay something to purchase that option. A significant part of financial mathematics is devoted to the question of what is the fair price I should pay for a call option.

Options originated in the commodities market, where farmers wanted to hedge their risks. Since then many types of options have been developed (options are also known as derivatives), and the amount of money invested in options has for the past several years exceeded the amount of money invested in stocks.

Introduction

In this chapter, we present a number of representation results for random fields defined on ℝ3, or on the sphere S2. We shall focus on random fields whose law verifies some invariance properties, namely isotropy (i.e., invariance with respect to rotations) and/or stationarity (i.e., invariance with respect to translations).

Section 5.2 contains facts that are implicitly used throughout the rest of this book, connecting strong isotropy to group representations and the Peter-Weyl Theorem – with special emphasis on spherical random fields, that is, random fields indexed by the unit sphere. In Sections 5.3 and 5.4 we focus on weak versions of stationarity and isotropy and the associated spectral representations. In particular, the content of Section 5.4 provides alternate proofs (with no group theory involved) of some of the main findings of Section 5.2. Note that Section 5.3 and Section 5.4 are not used in the rest of the book, so that some technical details have been skipped in order to keep the presentation as concise and to the point as possible. Adequate references are given below: in general, for a textbook treatment of stationary and isotropic fields on ℝm, one can consult Yadrenko [201], Adler [2], Ivanov and Leonenko [104], Leonenko [125], Adler and Taylor [3] and the references therein.

The Stochastic Peter-Weyl Theorem

General statements

We will now use notions and definitions that have been introduced in Chapter 2. Let G be a topological compact group, and let dg be the associated Haar measure with unit mass.

Introduction

In this chapter, we shall specialize the results of Chapter 2 to the compact group which is central for our analysis, namely the “special group of rotations” SO(3). The latter can be realized as the space of 3 × 3 real matrices A such that A′A = I3 (where I3 is the three-dimensional identity matrix) and det(A) = 1. In particular, we shall carry out an explicit construction for a complete set of irreducible representations of SO(3). To do so, we shall first establish a more general result, namely, we will provide (following a classical argument) a complete family of irreducible representations for the group SU(2); we will then recall a well-known relationship between SO(3) and SU(2) (i.e. that the latter “covers” the former twice, i.e. SO(3) ≃ SU(2)/{I2, -I2}, where the 2 × 2 identity matrix I2 is the identity element of SU(2)) and hence show that the representations of SO(3) are a subset of the representations of SU(2). We will then develop Fourier analysis on the sphere, largely by means of the Peter-Weyl Theorem discussed in the previous chapter. In particular, we shall prove that functions on the sphere can be identified with a subset of those on the group SO(3), so that their spectral representation will require only a subset of the matrix coefficients in the representation of the latter (more formally, we shall identify the sphere S2 as the quotient space SO(3)/SO(2)).