Financial Derivatives/Notions of Stochastic Calculus

Stochastic Process

A stochastic process ${\displaystyle X}$ is an indexed collection of random variables:

${\displaystyle X_{t}(\omega )}$

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

If we fix a ${\displaystyle t\in T}$ the stochastic process becomes the random variable:

${\displaystyle X_{t}=X_{t}(\omega )}$

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

Brownian Motion

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

- ${\displaystyle W_{0}=0}$

- It has independent, stationary increments. Let ${\displaystyle s\leq t}$, then: ${\displaystyle X_{t_{2}}-X_{t_{1}},\ldots ,X_{t_{n}}-X_{t_{n-1}}}$ are independent. And ${\displaystyle X_{t}-X_{s}=X_{t+h}-X_{s+h}\sim {\mathcal {N}}(0,t-s)}$

- ${\displaystyle W_{t}}$ is almost surely continuous