Financial Derivatives/Notions of Stochastic Calculus

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Stochastic Process[edit]

A stochastic process X is an indexed collection of random variables:

X_t(\omega)

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

If we fix a t\in T the stochastic process becomes the random variable:

X_t = X_t(\omega)

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

Brownian Motion[edit]

A stochastic process W_t(\omega) with t\in[0,\infty] is called a Wiener Procees (or Brownian Motion) if:

- W_0 = 0

- It has independent, stationary increments. Let s \leq t, then: X_{t_2} - X_{t_1}, \ldots, X_{t_n} - X_{t_{n-1}} are independent. And X_t - X_s = X_{t+h} - X_{s+h} \sim \mathcal{N}(0,t-s)

- W_t is almost surely continuous

References[edit]

Wikipedia on Stochastic Process Wikipedia on Wiener Process