Course notes for “Stochastic Process”

2014 Fall

**1. Motivation**

Brownian motion can be thought of a symmetric random walk where the jumps sizes are very small and where jumps occur very frequently.

- Each jump size are
- The time before two jumps are .

**1.1 What’s the mean and variance?**

Let

denote whether the i-th jump is to the right(+1) or to the left(-1), we have

with probability

and

respectively.

Thus,

and

.

Let denote the state of Markov Chain after n jumps, then

The denote the continuous Markov Chain after n jumps, then

Then we have

and .

Let , then

.

**2. Properties of Brownian Motion**

Answer: by the stationary property, we have

~

.

**3. Standard Brownian Motion (SBM)**

**4. Brownian Motion with Drift**

__Definition 1:__ Let be the standard Brownian motion. Let , then is *Brownian motion with drift* .

__Definition 2:__ . is Brownian Motion with drift and variance parameter if

- has stationary and independent increments
- ~

__Example:__ Let be Brownian motion with and drift . What is Pr{X(30) >0 | X(10) = -3}Pr{X(30) – X(20) >3 | X(10) =3 }Pr{X(30) – X(10) >3 }Pr{X(20) – X(0) >3 }Pr{X(20)>3 }Pr{N(2,80) > 3} = Pr{X(0,1) > frac{3-2}{sqrt{80}}1-Phi(frac{1}{4sqrt{5}})X(s) = x | X(t) = Bfrac{s}{t} cdot Bfrac{s}{t} cdot (t-s)s = t/2X(t)N(0, sigma^2 t)10 after 6 hours, what is the probability that the stack was above its starting value after 3 hours?

__Answer__:

This is a Brownian bridge process where

~

=

.

Thus

__Example 2:__ If a bicycle race between two competitors, Let denote the amount of time (in seconds) by which the racer that started in the insider position is ahead when 100t percent of the race has been completed, and suppose that , can be effectively modeled as a Brownian Motion process with variance parameter .

**6. First Passage Time**

Let denote the first time that standard Brownian motion hits level a (starting at X(0) = 0), assuming a >0, then we have

- : you know that at some time before t, the process hits a. From that point forward, you are just as likely to be above a as below a.
- : cannot be above a, because the first passage time to a is after t.

Thus,

,

.

Change variables: , we have .

By symmetry, we get .