Stochastic Process

Let $(\Omega ,F,P)$ be a probability space. A stochastic process is a collection $(X_t{)}_{t\in T}$ random variables $X_t:\Omega \to S$.

• $\Omega :=\text{sample space}$
  • flip a fair coin infinitely many times: $\Omega =\{H,T{\}}^{\mathbb{N}}$
• $F:=\text{collection of events}$
• $P:=\text{probability measure}$
• $S:=\text{state space}$
• $T:=\text{index set, often time}$
• $X:=T\times \Omega \to S,(t,\omega )\mapsto X_t(\omega )$