- A One-Dimensional Random Walk
- Multi-Dimensional Random Walks
- The Wiener Process (Brownian Motion)
- Properties of the Wiener Process
- Martingales
- The Langevin Equation

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

(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

(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

(11) |

(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 :

- They are continuous everywhere but differentiable nowhere.
- They are recurrent (i.e. the paths will return to the origin with probability 1) for and transient (i.e there is a finite probability different from 1 that the paths will return to the origin) for .
- The path is fractal in nature.

Equation (2.3) is now replaced by

(13) |

(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

(17) |

Hence, follows a normal distribution.

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

(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.

(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

(23) |

(24) |

(25) |

(26) |

(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

- If and is a sequence of independent centered random variables (i.e. and ), then is a martingale with respect to where and .
- If is a sequence of independent random variables with and for all , then is a martingale with respect to where .

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

(28) |

where is a stochastic process with zero mean and covariance

(30) |

(31) |

(32) |

(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

(34) |

(35) |

(36) |

As
, the particle attains equilibrium with its
surroundings. Hence, the velocity distribution should be Maxwellian

(37) |

The stochastic differential equation for the logarithm of the stock
price

(which, after changing variables to and setting (implying zero viscosity) and , is the same as the classical Langevin equation) Equation (2.37) can be readily solved to yield . We can formally solve equation (2.38) as above to obtain

(40) |

(41) |

(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.