Information Technology Reference
In-Depth Information
7.2.4
The Wiener Process Corresponds to a Diffusion PDE
It has been shown that the limit of the Wiener walk for n!1 is the Wiener process,
which corresponds to the partial differential equation of a diffusion [
56
]. For t>0,
function .x;t/ is defined as the probability density function (p.d.f.) of the Wiener
process, i.e.
Z
C1
EŒf.
w
.t// D
f.x/.x;t/
dx
(7.9)
1
As explained in Eq. (
7.6
), the p.d.f. .x;t/ is a Gaussian variable with mean value
equal to 0 and variance equal to
2
t which satisfies a diffusion p.d.e of the form
2
2
@
2
@
@t
D
1
(7.10)
@t
2
which is the simplest diffusion equation (heat equation). The generalization of the
Wiener process in an infinite dimensional space is the
Ornstein-Uhlenbeck
process,
where the joint probability density function is also Gaussian [
17
,
56
]. In that case
there are n Brownian particles, and each one performs a Wiener walk, given by
w
i
.t
k
/;i D 1; ;nof Eq. (
7.4
).
7.2.5
Stochastic Integrals
For the stochastic variable
w
.t/ the following integral is computed
I D
R
t
t
0
G.s/
dw
.s/
(7.11)
or
S
n
D
P
jD1
G.
j
/Œ
w
.t
j
/
w
.t
j1
/
(7.12)
where G.t/ is a piecewise continuous function. To compute the stochastic integral
the selection of sample points
j
is important. If
j
D t
j1
, then one has the Itô
calculus. If
j
D .t
j1
C t
j
/=2, then one has the Stratonovic calculus.
7.2.6
Ito's Stochastic Differential Equation
From the relation about the standard Wiener process given in Eq. (
7.7
), and
considering that the variance of noise is equal to 1, it holds that
EŒ.
w
.t C h/
w
.t//
2
D hEŒN.0;1/
2
D h
(7.13)
