Financial Derivatives/Notions of Stochastic Calculus
Stochastic Process 
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 Motion 
A stochastic process with is called a Wiener Procees (or Brownian Motion) if:
- It has independent, stationary increments. Let , then: are independent. And
- is almost surely continuous