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 Process (or Brownian Motion) if: - - It has independent, stationary increments. Let , then: are independent. And - is almost surely continuous References Wikipedia on Stochastic Process Wikipedia on Wiener Process