# Brownian Motion - University of lalley/Courses/385/ ¢ Exercise: Verify this. 1.3 Brownian

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Brownian Motion

1 Brownian motion: existence and first properties

1.1 Definition of the Wiener process

According to the De Moivre-Laplace theorem (the first and simplest case of the cen- tral limit theorem), the standard normal distribution arises as the limit of scaled and centered Binomial distributions, in the following sense. Let ξ1, ξ2, . . . be inde- pendent, identically distributed Rademacher random variables, that is, independent random variables with distribution

P{ξi = +1} = P{ξi = −1} = 1

2 , (1)

and for each n = 0, 1, 2, . . . let Sn = ∑n

i=1 ξi. The discrete-time stochastic process {Sn}n≥0 is the simple random walk on the integers. The De Moivre-Laplace theorem states that for each x ∈ R,

lim n→∞

P{Sn/ √ n ≤ x} = Φ(x) :=

∫ x −∞

ϕ1(y) dy where (2)

ϕt(y) = 1√ 2πt

e−y 2/2t (3)

is the normal (Gaussian) density with mean 0 and variance t.

The De Moivre-Laplace theorem has an important corollary: the family {ϕt}t≥0 of normal densities is closed under convolution. To see this, observe that for any 0 < t < 1 the sum Sn = S[nt] + (Sn − S[nt]) is obtained by adding two independent Rademacher sums; the De Moivre-Laplace theorem applies to each sum separately, and so by an elementary scaling we must have ϕ1 = ϕt ∗ ϕ1−t. More generally, for any s, t ≥ 0,

ϕs ∗ ϕt = ϕs+t. (4)

This law can, of course, be proved without reference to the central limit theo- rem, either by direct calculation (“completing the square”) or by Fourier transform.

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However, our argument suggests a “dynamical” interpretation of the equation (4) that the more direct proofs obscure. For any finite set of times 0 = t0 < t1 < · · · < tm

of the central limit theorem (5). Unfortunately, this approach requires a bit more technology (specifically, weak convergence theory for measures on metric space; see he book Weak Convergence of Probability Measures by P. Billingsley for details) than we have time for in this course. However, it has the advantage that it also explains, at least in part, why the Wiener process is useful in the modeling of natural processes. Many stochastic processes behave, at least for long stretches of time, like random walks with small but frequent jumps. The weak convergence theory shows that such processes will look, at least approximately, and on the appropriate time scale, like Brownian motion.

Notation and Terminology. A Brownian motion with initial point x is a stochastic process {Wt}t≥0 such that {Wt−x}t≥0 is a standard Brownian motion. Unless other- wise specified, Brownian motion means standard Brownian motion. To ease eyestrain, we will adopt the convention that whenever convenient the index t will be written as a functional argument instead of as a subscript, that is, W (t) = Wt.

1.2 Brownian motion and diffusion

The mathematical study of Brownian motion arose out of the recognition by Ein- stein that the random motion of molecules was responsible for the macroscopic phenomenon of diffusion. Thus, it should be no surprise that there are deep con- nections between the theory of Brownian motion and parabolic partial differential equations such as the heat and diffusion equations. At the root of the connection is the Gauss kernel, which is the transition probability function for Brownian motion:

P (Wt+s ∈ dy |Ws = x) ∆ = pt(x, y)dy =

1√ 2πt

exp{−(y − x)2/2t}dy. (6)

This equation follows directly from properties (3)–(4) in the definition of a standard Brownian motion, and the definition of the normal distribution. The function pt(y|x) = pt(x, y) is called the Gauss kernel, or sometimes the heat kernel. (In the parlance of the PDE folks, it is the fundamental solution of the heat equation). Here is why:

Theorem 2. Let f : R→ R be a continuous, bounded function. Then the unique (contin- uous) solution ut(x) to the initial value problem

∂u

∂t =

1

2

∂2u

∂x2 (7)

u0(x) = f(x) (8)

is given by

ut(x) = Ef(W x t ) =

∫ ∞ y=−∞

pt(x, y)f(y) dy. (9)

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Here W xt is a Brownian motion started at x.

The equation (7) is called the heat equation. That the PDE (7) has only one solution that satisfies the initial condition (8) follows from the maximum principle: see a PDE text for details. More important (for us) is that the solution is given by the expectation formula (9). To see that the right side of (9) actually does solve (7), take the partial derivatives in the PDE (7) under the integral in (9). You then see that the issue boils down to showing that

∂pt(x, y)

∂t =

1

2

∂2pt(x, y)

∂x2 . (10)

Exercise: Verify this.

1.3 Brownian motion in higher dimensions

Definition 2. A standard d−dimensional Brownian motion is an Rd−valued continuous- time stochastic process {Wt}t≥0 (i.e., a family of d−dimensional random vectors Wt indexed by the set of nonnegative real numbers t) with the following properties.

(A)’ W0 = 0. (B)’ With probability 1, the function t→ Wt is continuous in t. (C)’ The process {Wt}t≥0 has stationary, independent increments. (D)’ The increment Wt+s −Ws has the d−dimensional normal distribution with

mean vector 0 and covariance matrix tI .

The d−dimensional normal distribution with mean vector 0 and (positive definite) covariance matrix Σ is the Borel probability measure on Rd with density

ϕΣ(x) = ((2π) d det(Σ))−1/2 exp{−xTΣ−1x/2};

if Σ = tI then this is just the product of d one-dimensional Gaussian distributions with mean 0 and variance t. Thus, the existence of d−dimensional Brownian motion follows directly from the existence of 1−dimensional Brownian motion: if {W (i)}t≥0 are independent 1−dimensional Brownian motions then

Wt = (W (1) t ,W

(2) t , . . . ,W

(d) t )

is a d-dimensional Brownian motion. One of the important properties of the d−dimensional normal distribution with mean zero and covariance matrix tI pro- portional to the identity is its invariance under orthogonal transformations. This im- plies that if {Wt}t≥0 is a d−dimensional Brownian motion then for any orthogonal transformation U of Rd the process {UWt}t≥0 is also a d−dimensional Brownian motion.

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1.4 Symmetries and Scaling Laws

Proposition 1. Let {W (t)}t≥0 be a standard Brownian motion. Then each of the following processes is also a standard Brownian motion:

{−W (t)}t≥0 (11) {W (t+ s)−W (s)}t≥0 (12) {aW (t/a2)}t≥0 (13) {tW (1/t)}t≥0. (14)

Exercise: Prove this.

The scaling law (13) is especially important. It is often advisable, when con- fronted with a problem about Wiener processes, to begin by reflecting on how scaling might affect the answer. Consider,as a first example, the maximum and minimum random variables

M(t) := max{W (s) : 0 ≤ s ≤ t} and (15) M−(t) := min{W (s) : 0 ≤ s ≤ t}. (16)

These are well-defined, because the Wiener process has continuous paths, and continuous functions always attain their maximal and minimal values on compact intervals. Now observe that if the path W (s) is replaced by its reflection −W (s) then the maximum and the minimum are interchanged and negated. But since −W (s) is again a Wiener process, it follows that M(t) and −M−(t) have the same distribution:

M(t) D = −M−(t). (17)

Next, consider the implications of Brownian scaling. Fix a > 0, and define

W ∗(t) = aW (t/a2) and M∗(t) = max

0≤s≤t W ∗(s)

= max 0≤s≤t

aW (s/a2)

= aM(t/a2).

By the Brownian scaling property, W ∗(s) is a standard Brownian motion, and so the random variable M∗(t) has the same distribution as M(t). Therefore,

M(t) D = aM(t/a2). (18)

Exercise: Use Brownian scaling to deduce a scaling law for the first-passage time random variables τ(a) defined as follows:

τ(a) = min{t : W (t) = a} (19) or τ(a) =∞ on the event that the process W (t) never attains the value a.

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1.5 The Wiener Isometry

The existence theorem (Theorem 1) was first proved by N. Wiener around 1920. Simpler proofs have since been found, but Wiener’s argument contains the germ of an extremely useful insight, which is now known as the Wiener isometry (or, in some of the older literature, the Wiener integral). Following is an account of Wiener’s line of thought.

Suppose that Brownian motion exists, that is, suppose that on some probability space (Ω,F , P ) there is a centered Gaussian process {Wt}t∈[0,1] with covariance EWtWs = min(s, t). The random variables Wt are all elements of the space L2(P ) consisting of the real random variables defined on (Ω,F , P ) with finite second moments. This space is a Hilbert space with inner product 〈X, Y 〉 = (X, Y ).

Now consider the Hilbert space L2[0, 1], consisting of all real-valued square- integrable functions on the unit interval, with inner product

〈f, g〉 = ∫ 1

0

f(x)g(x) dx.

Indicator functions 1[0,t] of intervals [0, t] are elements of L2[0, 1], and obviously

〈1[0,t],1[0,s]〉 = min(s, t).

Thus, the indicators 1[0,t] have exactly the same inner products as do the random variables Wt in the Hilbert space L2(P ). Wiener’s key insight was that this identity between inner products implies that there is a linear isometry IW from L2[0, 1] into L2(P ) mapping each indicator 1[0,t] to the corre