STOCHASTIC PROCESSES 02
State Spaces
The state space is the set of all states that a stochastic process can occupy.
The main idea
If \(X_t\) describes the state at time \(t\), then
where \(S\) denotes the state space.
1. Finite state spaces
A process may have only finitely many states. If \(I_t\) counts infectious individuals in a population of size \(N\),
\[S=\{0,1,2,\ldots,N\}.\]2. Countably infinite state spaces
A queue length can take values
\[S=\{0,1,2,\ldots\}.\]The state space is infinite but countable.
3. Continuous state spaces
A process such as Brownian motion can take real-valued states:
\[S=\mathbb{R}.\]4. Multidimensional states
A state may contain several quantities. An epidemic state can be
\[X_t=(S_t,I_t,R_t).\]The state space then consists of all admissible combinations.
5. Boundary states
Some states have special behaviour. In an epidemic model, \(I=0\) may be absorbing because infection cannot restart without an external introduction.
6. Why the state must contain enough information
The state should describe everything needed by the model to determine future transition probabilities or rates. This becomes especially important for Markov models.