Stochastic process:

Brownian motion :

- Continuous
- Independent increments
- Normal distribution of increments, i.e.

Symmetric random walk (SRW):

- with probability and respectively
- SRW is and
- Scaled SRW:
- Limit: Let , then

Covariance of BM, assumed ,

Therefore, we have

Geometric BM:

First passage time problem:

- Let be the
*running maximum*of stochastic process - Let be the first passage time
- is equivalent to

Reflection principle of Brownian motion :

To obtain the p.d.f. of , we can consider the m.g.f. .

- Consider the trail of a Brownian motion that , , for , , then it has the same probability as that , , , just as the path of the B.M. reflected about .
- .
- .
- .

Stochastic integral:

Ito integral:

- is a -adapted, square-integrable stochastic process
- It is a martingale
- .

Ito formula:

Example: Solving

Ito formula in general form:

where , and

Multidimensional version: