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: