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Wiener processƒ 227

Norbert Wiener, 1920:

W

is a t

Wiener process,

i.p.

W

t

is a random (stochastic)

real-valued continuous fu

nction (process) on [0,



) such that:

ƒ

W

t=0

= 0,

ƒ

dW

= Wt

t+dt

-W

~ N(0,dt), andt

ƒ

if the intervals [t1, t2] and [u1, u2] do

not

overlap, then the increments dW

= Wt

t2

W

t1

and dW

u

= W

u2

-W

u1

are independent!

ƒ

One realization of a Wiener process

ƒ

Some implied properties

ƒ

W is nowhere differentiable

due to its jaggedness which is a result of the

independent increments

ƒ

Since each increment of

W is normal distributed

W itself is normal distributed

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