DES - Definition of 'Stochastic Process'

Stochastic Process:

A stochastic process $\left\{X(t)\right\}$ is a collection of random variables $X(t)\in \mathcal{X}$, indexed by time $t\in \mathcal{T}$.

• If the time set $\mathcal{T}$ is discrete, ex.: $\mathcal{T}=\{0,1,2,\dots \}$ the stochastic process is in discrete time

• If the time set $\mathcal{T}$ is continuous, ex.: $\mathcal{T}={\mathbb{R}}^{\ge 0}$ the stochastic process is in continuous time

• If the time set $\mathcal{X}$ is discrete, the stochastic process is called chain

• A realization of the stochastic process $\left\{X(t)\right\}$ is denoted $x(t)$

• A stochastic process is specified by all possible joint distributions of its random variables:

$$\begin{array}{rl}& P(X(t_1)\le x_1,X(t_2)\le x_2,\dots ,X(t_h)\le x_h)\\ & \forall 0\le t_1\le t_2\le \cdots \le t_h,\\ & \forall 0\le x_1\le x_2\le \cdots \le x_h,\\ & \forall h=1,2,3,\dots \end{array}$$

• A stochastic process is called (strongly) stationary if these distributions are invariant with respect to a shift in time:

$$\begin{array}{rl}& P(X(t_1+\tau )\le x_1,X(t_2+\tau )\le x_2,\dots ,X(t_h+\tau )\le x_h,)=\\ & =P(X(t_1)\le x_1,X(t_2)\le x_2,\dots ,X(t_h)\le x_h,)\\ & \forall 0\le t_1\le t_2\le \cdots \le t_h,\\ & \forall 0\le x_1\le x_2\le \cdots \le x_h,\\ & \forall h=1,2,3,\dots \\ & \forall \tau >0\end{array}$$

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Observation:

Notice that strong stationarity implies all the random variables $X(t)$ are identically distributed

Suppose: $h=1$, then:

$$\begin{array}{rl}& P(X(t_1+\tau ))=P(X(t_1)\le x_1)\\ & \forall 0\le t_1\le t_2\le \cdots \le t_n,\\ & \forall 0\le x_1\le x_2\le \cdots \le x_n,\\ & \forall h=1,2,3,\dots \end{array}$$

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~Example: continuous state

$\mathcal{T}$ continuous $\mathcal{X}$ continuous

~Example: discrete state

$\mathcal{T}$ continuous $\mathcal{X}$ discrete