Mathematical Modeling in Finance with Stochastic Processes

156 CHAPTER 5. BROWNIAN MOTION

Mathematical Ideas

The question is “How should we set up the limiting process so that we can
make a continuous time limit of the discrete time random walk?” First
we consider a discovery approach to this question by asking what should
be the limiting process which assures us that we can visualize the limiting
process. Next we take a probabilistic view using the Central Limit Theorem
to justify the limiting process to pass from a discrete probability distribution
to a probability density function. Finally, we consider the limiting process
derived from passing from the difference equation from first-step analysis to
a differential equation.

Visualizing Limits of Random Walks

The Random Walk

Consider a random walk starting at the origin. Thenth step takes the walker
to the positionTn =Y 1 +···+Yn, the sum of n independent, identically
distributed Bernoulli random variablesYiassuming the values +1, and− 1
with probabilitiespandqrespectively. Then recall that the mean of a sum
of random variables is the sum of the means:

E[Tn] = (p−q)n

and the variance of a sum ofindependentrandom variables is the sum of the
variances:

Var [Tn] = 4pqn.

Trying to use the mean to derive the limit

Now suppose we want to display a motion picture of the random walk moving
left and right along thex-axis. This would be a motion picture of the “phase
space” diagram of the random walk. Suppose we want the motion picture to
display 1 million steps and be a reasonable length of time, say 1000 seconds,
between 16 and 17 minutes. This fixes the time scale at a rate of one step
per millisecond. What should be the window in the screen in order to get
a good sense of the random walk? For this question, we use a fixed unit of
measurement, say centimeters, for the width of the screen and the individual
steps. Letδbe the length of the steps. To find the window to display the