Brownian motion :
- 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
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 .
- is a -adapted, square-integrable stochastic process
- It is a martingale
Ito formula in general form:
where , and