In this chapter, I will present the theory of stochastic processes in an elementary manner sufficient for understanding the theory presented in the following chapters. A reader interested in a more rigorous approach could consult Ross[4]. This chapter mostly follows Roepstorff[5] which has a more physical description of stochastic processes.
This process is both homogeneous (since the transition probability is only dependent on the distance between the initial and final points) and isotropic (the transition probability is independent of the direction of movement).
The random walk can be taken to be a Markov chain with a transition
matrix (for one time step) where the element
stands for
the probability that an ant with initial position
will end up at
. For
time steps, the accumulated transition probability is
given by
.
It can be easily seen that the following difference equation holds
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(6) |
Now, if we re-scale so that the step size is and the time step is
, the above becomes
In the limit
,
becomes
(now, of course,
and
are continuous) and
becomes
. Hence, in this limit, we get the following
equation for the one-dimensional random walk
Equation 2.4 is, of course, the diffusion equation which
should not come as too much of a surprise as diffusion is the result
of the random movement of molecules. The reader should also be
familiar with the fact that the solution of the diffusion equation
with the initial condition
is given by the
Gaussian
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(9) |
We have not handled the transition from discrete variables to
continuous variables rigorously but the steps above should be
intuitively reasonable. To do the above rigorously, we only have to
identify the discrete probability with
.
Throughout the above discussion, we have assumed that the probability
of the ant moving left or right is the same (i.e. we have assumed that
the walk is isotropic). This type of random walk is called the simple
random walk. To generalize the above discussion, we have to change
eqn. (2.1) to
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(11) |
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(12) |
The resulting paths have certain interesting geometrical properties. While these are not important for the purposes of this thesis, they are interesting in their own right. A few of them are described below :
Equation (2.3) is now replaced by
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(13) |
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(14) |
It is instructive to note that the mean square displacement
For simplicity, we will assume
in the rest of this
discussion. This can always be achieved by rescaling the time and
space coordinates.
Consider a particle which was initially at the origin performing a
-dimensional simple random walk. In other words,
at time
. The position of the particle at
time
can be considered as a random variable
(the term random vector might be more appropriate as it emphasis the fact that
has several components). Now, if we can find the probabilities
for all
with non-zero measure,
we would have described the path followed by the particle completely.
We have actually found the answer to this in eqn. (2.11). To
make this clearer, we can express the answer in the following manner
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(17) |
A stochastic process is a map , where
ranges over
some interval (
in this case). To properly define a
stochastic process, we need to be able to determine the probabilities
of general events. Before we can do this, it is necessary to consider
compound events of the form ``
'', where
and to devise rules that determine their
probability.
We denote the probability of a compound event by
Varying
we get the joint distribution of
the random variables
. This is
also called the distribution of the process with base
and may be
abbreviated as
where
. The distribution is said to be
finite-dimensional (of order
) since the base is finite (has
elements).
The stochastic process is said to be the Wiener
process
if the finite-dimensional
distributions are of the form
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(20) |
Hence, the particle starts at the origin (in other words,
with certainty).
We can rewrite eqn. (2.15) as
The fact that the right hand side of (2.17) is a product tells us that the Wiener process is a Markov (memory-independent) process. This is reassuring as the present state then contains all the information that is relevant for the future which we have seen is true for an efficient market.
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(22) |
For the rest of this discussion, we will restrict ourselves to the
case . The generalization to higher dimensions is
self-evident.
We will be particularly interested in the following expected value
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(23) |
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(24) |
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(25) |
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(26) |
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(27) |
A precise meaning can be given to by considering the concept of
generalized stochastic processes. A good discussion of this concept
can be found in Roepstorff[5].
A discrete stochastic process
is said to be a martingale
with respect to a process
if, for all
,
is called a sub-martingale with respect to
if, for
all
,
is a function of
and
where
.
is called a super-martingale with respect to
if, for
all
,
is a function of
and
where
.
While a martingale describes a fair game, the sub-martingales and
super-martingales describe favourable and unfavourable games
respectively. If is a martingale,
, while
if
is a sub-martingale and
if
is a super-martingale.
Some simple examples of martingales are
Martingales are extremely important in finance due to the concept of
risk-neutral valuation. This is due to the fact that the expected
growth rate of all securities in a risk-neutral world is the risk-free
interest rate. Hence, is a martingale for all securities
in a risk-neutral world. This is why risk-neutral valuation
approach is also referred to as using an equivalent martingale
measure.
Martingales have several interesting properties and several important limit theorems about them can be proven. These are beyond the scope of this thesis and the interested reader should consult a good textbook on stochastic processes such as Ross[4].
Langevin considered the equation of motion of a particle in a fluid
which is classically given by
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(28) |
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(30) |
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(31) |
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(32) |
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(33) |
We now solve the Langevin equation formally and check that the
solution gives us the same expected value and variance as the solution
above. The formal solution for the Langevin equation gives
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(34) |
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(35) |
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(36) |
As
, the particle attains equilibrium with its
surroundings. Hence, the velocity distribution should be Maxwellian
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(37) |
The stochastic differential equation for the logarithm of the stock
price
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(40) |
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(41) |
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(42) |
The main importance of the Langevin equation is that it gives us a different way of considering stochastic processes. They can also sometimes suggest better methods of solving stochastic differential equations. Mathematically, the two approaches are, of course, equivalent.