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Stochastic Process

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Definition

StochasticProcessesProbability

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $\mathbb{T}=\mathbb{N}$ . A stochastic process on $(\Omega,\mathcal{F},P)$ indexed by $\mathbb{T}$ is a family $(X_{t})_{t\in\mathbb{T}}$ of RVs on $(\Omega,\mathcal{F},P)$ i.e. $\forall t\in\mathbb{T}:\forall B\in\mathcal{B}(\mathbb{R}):X_{t}^{-1}(B)\in\mathcal{F}$ We define a stochastic process as a sequence of random variables, $\{ X_n(\omega) : n\in\mathbb{N}\}$ , such that: 1. For each $n\ge 0$ , $X_n:\Omega\to\mathbb{R}$ 2. For each $\omega\in \Omega$ , $\{X_n(\omega):n\in\mathbb{N}\}$ is called a trajectory 3. Since $\{X_n(\omega):n\in\mathbb{N}\}$ depends on $\omega$ , the trajectory is random.