Try the Free Math Solver or Scroll down to Resources!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


INTRODUCTION TO PROBABILITY

Lecture 37. Application to Markov processes. The Wiener process.

Theorem 37.1. Let be a probability measure on the space (X, ), and let
P( s, x, t, C) be a function of the times s, t, 0≤ s≤t, of x ∈X, and of a set C ∈ ,
such that for fixed s, t; and x it is a probability measure as a function of C; for fixed
s, t, and C it is measurable as a function of x; P (t, x, t, C) = (C); and this function
satisfies the Chapman - Kolmogorov equation:

Then the finite-dimensional distributions defined, for 0 ≤ t1≤ t2 ... ≤tn,
by


satisfy the consistency conditions.

The proof is very simple: it could seem that the conditions imposed on P (s, x, t, C)
were designed specially so that the proof is easy. The condition (36.9n+1) is satisfied
because P (tn, xn, tn+1, X) = 1, and conditions (9i), 1 ≤ i ≤ n, because of the Chapman {
Kolmogorov equality

So if the space (X, ) is not extremely ugly (e. g. a non-Borel set with I don't
know what as the σ-algebra in it), there exists a stochastic process with the prescribed
finite-dimensional distributions, and by Theorem 34.1 it is a Markov process with initial
distribution ν and transition function P(  , , ,). In particular, there is a Markov
process corresponding to the transition function you invented solving Problem 54 . Can
you say anything about this Markov process of yours; apart from its finite-dimensional
distributions?

Another example:

Example 37.1. Take The transition density

(as a function of y, it is the density of the normal distribution with parameters (x, t-s))
satisfies the Chapman -Kolmogorov equation:

One of the ways to check it is to collect all terms with y2, with y, and with no
y in the quadratic function being the exponent in the integrand, and use the fact that

Another way: For fixed x, the integrand in (37.3) is, as a function of y, z, the den-
sity of the two-dimensional normal distribution with parameters
(check it!); the integral with respect to y, as a function of z, represents the density of the
second coordinate: that of the normal distribution with parameters (x, u- s).

A third way: the left-hand side integral in (37.3) is the convolution formula for the
probability density of the sum of two independent one-dimensional random variables
and , where has the normal distribution with parameters (x, t - s), and the second,
, with parameters (0, u- t). The characteristic function of the
sum is equal to the product of their characteristic functions:
the characteristic function corresponding to
the normal density in the right-hand side of (37.3) (you see, I am running out of letters).

Problem A Check that the density satisfies the
Chapman -Kolmogorov equation.

A stochastic process is a function of two arguments: the "time"
and the sample point  ω∈Ω.

If we fix t ∈ T, the function of the second argument is a random variable.
If we fix ω, is just a function of "time". We call this function a trajectory (a
sample function) of our stochastic process.

A stochastic process with values in is called a Wiener process
if its finite-dimensional distributions are given by formula (37.1) with

and all its trajectories are continuous: is continuous in t for every ω∈Ω

We have proved that it is possible to satisfy the first part of the definition { that
about the finite-dimensional distributions; but we know nothing about the second part {
about all trajectories being continuous. The credit for this belongs to N.Wiener.

The Wiener process is the mathematical model for the physical phenomenon of Brownian motion:
the process of motion of a light particle in a fluid under the influence of chaotic impulses from molecules
hitting it. That is, the Brownian motion is three-dimensional (or two-dimensional if we want to model what
we see under the microscope), while we consider the one-dimensional Wiener process; more accurate would
be saying that the Wiener process is the mathematical model of one coordinate of the physical Brownian
motion.

The stochastic process having the given finite-dimensional distributions
(no matter what these distributions are) constructed in the proof of Kolmogorov's Theorem

(Theorem 35.2) clearly doesn't have all trajectories continuous: the trajectories =
are all functions belonging to (all real-valued functions on the interval
[0, 1)), and we know that there are discontinuous functions. We cannot state either that
almost all functions are continuous: it can be proved easily that
makes no sense: the set does not belong to the

We may hope that there is, on the same probability space , another
stochastic process, t ∈ [0, ∞), with the same finite-dimensional distributions, but with
continuous trajectories.

Theorem 37.2. If and are two random functions; and
for every t ∈ T, then the random functions and have the same
finite-dimensional distributions.

Proof. We have to prove that for every natural n, for every t1, ..., tn ∈ T, and for
every C∈

The symmetric difference of the events and is
clearly a subset of the event and the probability of this union
is not greater than

In the next lecture we'll have a bigger theorem that will help us to prove the existence
of a Wiener process.