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# 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.