Financial Derivatives/Notions of Stochastic Calculus

Stochastic Process

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

$X_t(\omega )$

Where $\omega \in \mathrm{\Omega }$ 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 $\omega \in \mathrm{\Omega }$ we obtain a deterministic function of time: a realization or sample path of the process.

Brownian Motion

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

- $W_0=0$

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

- $W_t$ is almost surely continuous

References

Wikipedia on Stochastic Process Wikipedia on Wiener Process