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

References

Wikipedia on Stochastic Process Wikipedia on Wiener Process