Financial Derivatives/Notions of Stochastic Calculus

Stochastic ProcessEdit

A stochastic process   is an indexed collection of random variables:

 

Where   our sample space, and   is the index of the process which may be either discrete or continuous. Typically, in finance,   is an interval   and we deal with a continuous process. In this text we interpret   as the time.

If we fix a   the stochastic process becomes the random variable:

 

On the other hand, if we fix the outcome of our random experiment to   we obtain a deterministic function of time: a realization or sample path of the process.

Brownian MotionEdit

A stochastic process   with   is called a Wiener Process (or Brownian Motion) if:

-  

- It has independent, stationary increments. Let  , then:   are independent. And  

-   is almost surely continuous

ReferencesEdit

Wikipedia on Stochastic Process Wikipedia on Wiener Process