Stochastic Processes
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:
- Discrete time and discrete state processes:
- Examples: Markov Chains, branching processes, Random Walks.
- Discrete time and continuous state processes:
- Examples: ARMA models, GARCH models.
- Continuous time and discrete state processes:
- Examples: Poisson processes, Birth-Death Processes.
- Continuous time and continuous state processes:
3 Stochastic Process as a Random Function
A stochastic process can be viewed as a single random element taking values in a space of functions. The map \(\omega\mapsto(t\mapsto X_t(\omega))\) sends each outcome to a path in the space of functions \(T\to\mathbb{R}^n\), and the process is determined in law by its finite-dimensional distributions, the joint laws of \((X_{t_1},\dots,X_{t_k})\) for all finite choices of times \(t_1,\dots,t_k\in T\). For continuous time, properties of paths such as continuity are not determined by these alone; see Kolmogorov's Continuity Theorem. The information revealed up to time \(t\) is modelled by a filtration, see Filtrations and Adapted Processes, and random times that depend only on this information are Stopping Times.