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5.2 Stochastic theory
Slide : [
mean & variance 
normal 
uniform 
Wiener 
Ito 
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This section is intended as a very short introduction into the stochastic
theory, which provides the mathematical foundation behind the Monte Carlo
method; check Sczepessy [28], Kloeden [22] and Van
Kampen [20] for complete courses on the subject.
To begin, remember the definition of an expected or mean value
and the variance
where
is the density distribution function of a stochastic
variable
.
The notation
refers to the set of Gaussian
or normally distributed stochastic variable
with a density
distribution function

(4) 
where
is the mean and
is the variance.
Along the same lines,
stands for the set of uniformly
distributed random numbers in the interval
.
Now consider a class of stochastic processes that play a central role in
the Monte Carlo method:
is called a Wiener process (or in an
equivalent manner Brownian motion) at a time
if and only if

,
where
is the set of normal distributed random numbers.

,
where
denotes expected value,
and
, i.e. the Wiener increment
is independent
of the past.
It turns out that the distribution of Wiener increments
is essentially the same function as the Green's function of the diffusion
equation (1.3.2#eq.3).
The differential calculus of stochastic processes (or
Itô calculus) involves new properties that are fundamentally
different from the ordinary (or Riemann) calculus. The reason can be
tracked down to the preferred direction of time
in the
Itô integral, which is defined as the limit of the explicit
(forward Euler) discretization

(5) 
for any sequence
.
Note that the circle
here stands for Itô differential and states
that
is independent of
; also, an implicit discretization
in (5.2#eq.5) would give a fundamentally different result.
Define a property
that is a sum of a Riemann and an Itô integral

(6) 
For an infinitely short time interval, this leads to the stochastic
differential equation

(7) 
which reduces to an ordinary differential equation in the absence of the
stochastic component

(8) 
Stochastic differential equations such as (5.2#eq.7) are used in the
MonteCarlo method to describe the trajectories or orbits of particles.
SYLLABUS Previous: 5.1 Monte Carlo integration
Up: 5 MONTECARLO METHOD
Next: 5.3 Particle orbits
