Stochastic Process

Definition (Stochastic Process)

A Stochastic Process is a family of random variables $X_t\in \Omega$ indexed by $t$ where $S\subseteq {\mathbb{R}}^d$ is the state space.

Typically, $t\in [0,\infty )$ or $t\in [a,b]$ or $t\in \{0,1,2,3,\dots \}$ etc.

Definition (Discrete Time Stochastic Process)

A discrete time stochastic process is a sequence of random variables $X_n\in S$ for $n=0,1,2,3,\dots$ (or excluding $0$). If $S$ is finite or countably infinite, then the stochastic process is a discrete state process.

Definition (Joint Distribution of Stochastic Processes)

The Joint Distribution of a stochastic process is

$$\mathbb{P}(X_0=i_0,X_1=i_1,\dots ,X_n=i_n)$$

where $i_j\in S$ for $1\le j\le n$.