A stochastic process \(X_t\) is a parameterized collection of random variables \((X_t)_{t\in T}\) defined on a [[knowledge-mathematics-analysis-probability-theory-probability-spaces|probability space]] \((\Omega, \mathcal{F}, \mathbb{P})\) and assuming values in \(\mathbb{R}^n\).
- The parameter space \(T\) usually represents time, either as discrete increments \(\{0,1,2,...,T\}\) or as a continuous interval \([0,\infty)\).
- Note that for each \(t\in T\) fixed we have a random variable \(\omega\rightarrow X_t(\omega)\) for \(\omega\in\Omega\). On the other hand, fixing \(\omega\in\Omega\) we can consider a path of \(X_t\) as the function \(t\rightarrow X_t(\omega)\) for \(t\in T\).
- It is useful to think of \(t\) as time and \(\omega\) as a realization of the process.
0.0.0.1 Types of Stochastic Process
- [[wiener-process-brownian-motion]]
- Geometric Brownian Motion