**1. Motivation**

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

**1.1 What’s the mean and variance?**

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**

**(1)****(2)****(3) X(t) has independent increments**

i.e., is independent of assuming the intervals of and are disjoint.- (4) X(t) has stationary increments

i.e., has the same distribution as if .

**Example**: What’s the distribution of ?

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

- SBM ~
- Let , then .

**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__:

__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.

Change variables: , we have .

By symmetry, we get .