Stochastic Process

1 What is a Stochastic Process?

A stochastic process \(X_t\) is a parameterized collection of random variables \((X_t)_{t\in T}\) defined on a 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.

2 Types of Stochastic Process

Broadly speaking, stochastic processes can be classified into 4 groups:

1. Discrete time and discrete state processes:
   • Examples: Markov chains, branching processes, random walks.
2. Discrete time and continuous state processes:
   • Examples: ARMA models, GARCH models.
3. Continuous time and discrete state processes:
   • Examples: Poisson processes, birth-death processes.
4. Continuous time and continuous state processes:
   • Examples: Brownian motion, Ornstein-Uhlenbeck processes, geometric Brownian motion.

3 Stochastic Process as a Random Function

Backlinks

• Cylinder Sets
• Probability Theory
• Real Analysis
• Stochastic Control
• Stochastic Games