Mathematical Modeling in Finance with Stochastic Processes

5.2 The Definition of Brownian Motion and the Wiener Process

Problems to Work for Understanding

1. Consider a random walk with a step to right having probabilitypand
   a step to the left having probability q. The step length isδ. The
   walk is takingrsteps per minute. What is the rate of change of the
   expected final position and the rate of change of the variance? What
   must we require on the quantitiesp,q,randdeltain order to see the
   entire random walk with more and more steps at a fixed size in a fixed
   amount of time?


2. Verify the limit taking to show that

vk,n∼(1/(

√

2 πDt)) exp(−[x−ct]^2 /(2Dt)).

3. Show that

v(t,x) = (1/(

√

2 πDt)) exp(−[x−ct]^2 /(2Dt))

is a solution of

∂v(t,x)

∂t

=−c

∂v(t,x)

∂x

+ (1/2)D

∂^2 v(t,x)

∂x^2

by substitution.

Outside Readings and Links:

1. Brownian Motion in Biology. A simulation of a random walk of a sugar
   molecule in a cell.


2. Virtual Laboratories in Probability and Statistics. Search the page for
   Random Walk experiment.

5.2 The Definition of Brownian Motion and

the Wiener Process

Rating

Mathematically Mature: may contain mathematics beyond calculus with
proofs.