STOCHASTIC PROCESSES 03
Discrete and Continuous Time
A stochastic process can be indexed at separate time steps or at every point in continuous time.
Discrete time
A discrete-time process is observed at times such as
\[t_0,t_1,t_2,\ldots\]often with a fixed step \(\Delta t\):
\[t_n=n\Delta t.\]We may write the process as \(X_0,X_1,X_2,\ldots\).
1. Continuous time
A continuous-time process is defined for every time in an interval:
\[\boxed{\{X_t:t\ge0\}}.\]Events need not wait for a fixed observation step.
2. Time type and state type are different
A process can have discrete time and discrete states, continuous time and discrete states, or continuous time and continuous states. These are separate modelling choices.
3. DTMC
A discrete-time Markov chain changes or remains in its state at fixed steps. Transition probabilities describe what happens at the next step.
4. CTMC
A continuous-time Markov chain occupies discrete states but jumps at random continuous times. Transition rates describe how quickly jumps occur.
5. Brownian motion
Brownian motion has continuous time and a continuous state space. Its path changes continuously rather than through discrete jumps.