Chapter 64

Wiener ProcessesA real valued random variable X is normally distributed with mean 0 and variance σ2 if

P(X ∈ A) =1√

2πσ

∫A

e−12

x2

σ2 dx

Consider the characteristic function. By definition it is

φ X (λ )≡∫R

eiλxdλ X (x)

where λ X is the distribution measure for this random variable. Thus the characteristicfunction of this random variable is

1√2πσ

∫∞

−∞

eiλxe−1

2σ2 x2dx

One can then show through routine arguments that this equals exp(− 1

2 σλ2)

.

64.1 Real Wiener ProcessesHere is the definition of a Wiener process.

Definition 64.1.1 Let W (t) be a stochastic process which has the properties that whenevert1 < t2 < · · ·< tm, the increments {W (ti)−W (ti−1)} are independent and whenever s< t, itfollows W (t)−W (s) is normally distributed with variance t−s and mean 0. Also t→W (t)is Holder continuous with every exponent γ < 1/2 and W (0) = 0. This is called a Wienerprocess.

Do Wiener processes exist? Yes, they do. First here is a simple lemma which has reallybeen done before. It depends on the Kolmogorov extension theorem, Theorem 59.2.3 onPage 1862.

Lemma 64.1.2 There exists a sequence, {ξ k}∞

k=1 of random variables such that

L (ξ k) = N (0,1)

and {ξ k}∞

k=1 is independent.

Proof: Let i1 < i2 · · ·< in be positive integers and define

µ i1···in (F1×·· ·×Fn)≡1(√2π)n

∫F1×···×Fn

e−|x|2/2dx.

Then for the index set equal to N the measures satisfy the necessary consistency conditionfor the Kolmogorov theorem. Therefore, there exists a probability space, (Ω,P,F ) and

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