Financial Derivatives/Notions of Stochastic Calculus

Stochastic Process
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
A stochastic process $$W_t(\omega)$$ with $$t\in[0,\infty]$$ is called a Wiener Process (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