and 
approach 0.
This time we will give a (first) derivation of the Fokker-Planck equation, which governs the evolution of the probability density function of a random variable-valued function X(t) that satisfies a "first-order stochastic differential equation". The variable evolves according to what in the literature is called a Wiener process, named after Norbert Wiener, who developed a great deal of the theory of stochastic processes during the first half of this century.
We shall begin with a generalized version of random walk (an
"unbalanced" version where the probability of moving in one direction is
not necessarily the same as that of moving in the other). Then we shall let
both the space and time increments shrink in such a way that we can
make sense of the limit as both and approach 0.
Generalized random walk
We consider a generalized one-dimensional random walk (on the integers) beginning at X=0. We fix a parameter p, and let p be the probability of taking a step to the right, so that q=1-p is the probability of taking a step to the left, at any (integer-valued) time t. In symbols, this is:
Since we are given that X(0)=0, we know that the probability function of
X(n) will be essentially binomially distributed (with parameters n and p,
except that at time n, x will go from -n to +n, skipping every other integer
as discussed earlier). We will set 
. Then
for 
and
otherwise.
For use later, we calculate the expected value and variance of X(n). First,
(it's an exercise to check the algebra; but the result is intuitive). Also,
(another exercise!). Also, note that we could have thought of X(n) as the sum of n independent binomially distributed random variables each having expected value p-q and variance 4pq. The same results follow from this observation.
Now, we let the space and time steps be different from 1. First, the time
steps (that's easier). We will take time steps of length . This
will have little effect on our formulas, except everywhere we will have to
note that the number n of time steps needed to get from time 0 to time t
is
For X, we change our notation so that each step to the left or right has
length instead of length 1. This has the effect of scaling X(t)
by the factor . Multiplying a random variable by a
constant has the effect of multiplying its expected value by the same
constant, and its variance by the square of the constant.
Therefore, the mean and variance of the total displacement in time t are:
We want both of these quantities to remain finite for all fixed t, since we don't want our process to have a positive probability of zooming off to infinity in finite time, and we also want Var(X(t)) to be non-zero. Otherwise the process will be completely deterministic.
We need to examine what we have control over in order to achieve these
ends. First off, we don't expect to let and to go to
zero in any arbitrary manner. In fact, it seems clear that in order for the
variance to be finite and non-zero, we should insist that be
roughly the same order of magnitude as 
. This is because we
don't seem to have control over anything else in the expression for the
variance. It is reasonable that p and q could vary with , but we
probably don't want their product to approach zero (that would force our
process to move always to the left or always to the right), and it can't
approach infinity (since the maximum value of pq is 1/4). So we are forced
to make the assumption that 
remain bounded
as they both go to zero. In fact, we will go one step further. We will insist
that
for some (necessarily positive) constant D.
Now, we are in a bind with the expected value. Under assumption A1, the
expected value will go to infinity as 
, and the only
parameter we have to control it is p-q. We need p-q to have the same order
of magnitude as in order to keep E(X(t)) bounded. Since p=q
when p=q=1/2, we will make the assumption that
for some real (positive, negative or zero) constant c, which entails
The reason for the choice of the parameters will become apparent. D is called the diffusion coefficient, and c is called the drift coefficient.
Note that as and approach zero, the values
specify
values of the function f at more and more points in the plane. In fact, the
points at which f is defined become more and more dense in the plane as
and approach zero. This means that given any point
in the plane, we can find a sequence of points at which such an f is
defined, that approach our given point as and
approach zero.
We would like to be able to assert the existence of the limiting function, and that it expresses how the probability of a continuous random variable evolves with time. We will not quite do this, but, assuming the existence of the limiting function, we will derive a partial differential equation that it must satisfy. Then we shall use a probabilistic argument to obtain a candidate limiting function. Finally, we will check that the candidate limiting function is a solution of the partial differential equation. This isn't quite a proof of the result we would like, but it touches on the important steps in the argument.
We begin with the derivation of the partial differential equation. To do
this, we note that for the original random walk, it was true that 
satisfies the following difference equation:
This is just a specification of the matrix of the (infinite discrete) Markov
process that defines the random walk. When we pass into the ,
realm, and use f(x,t) notation instead of 
, this equation becomes:
Next, we need Taylor's expansion for a function of two
variables. As we do the expansion, we must keep in mind that we are going
to let 
, and as we do this, we must account for all terms of
order up to and including that of . Thus, we must account for
both and terms (but 
and 
both go to zero faster than ).
We expand each of the three expressions in equation (*) around the point
(x,t), keeping terms up to order and . We get:
We substitute these three equations into (*), recalling that p+q=1, and get
(where
all the partial derivatives are evaluated at (x,t)). Finally, we divide
through by , and recall that 
, and 
to arrive at the equation:
This last partial differential equation is called the Fokker-Planck equation.
To find a solution of the Fokker-Planck equation, we use a probabilistic
argument to find the limit of the random walk distribution as
and approach 0. The key step in this argument uses the fact that
the binomial distribution with parameters n and p approaches the normal
distribution with expected value np and variance npq as n approaches
infinity. In terms of our assumptions about , , p and q,
we see that at time t, the binomial distribution has mean
and variance
which approaches 2Dt as . Therefore, we expect X(t) in the
limit to be normally distributed with mean 2ct and variance 2Dt. Thus, the
probability that 
is
Therefore the pdf of X(t) is the derivative of this with respect to x, which yields
Homework
1. Verify that the expected value and variance of 
are as
claimed in the text.
2. Verify that the function f(x,t) obtained at the end of the section satisfies the Fokker-Planck equation.