STOCHASTIC PROCESSES 10
Transition Rates
A transition rate measures how quickly a continuous-time Markov process tends to make a particular jump.
Rate is not probability
A rate \(q_{ij}\) has units of inverse time. It is not restricted to the interval \([0,1]\).
For sufficiently small \(\Delta t\),
The product is dimensionless and approximates a probability.
1. Infinitesimal definition
More precisely, for \(i\ne j\),
\[q_{ij}=\lim_{h\downarrow0}\frac{P(X_{t+h}=j\mid X_t=i)}{h}.\]2. Total rate
If several transitions can leave state \(i\), the total exit rate is
\[a_i=\sum_{j\ne i}q_{ij}.\]3. Small-time probabilities
\[P(\text{one particular jump})=q_{ij}h+o(h),\] \[P(\text{no jump})=1-a_i h+o(h).\]The probability of two or more jumps is of smaller order than \(h\).
4. State-dependent rates
Rates can depend on the current state. In an epidemic model, an infection rate may be
\[b(i)=\beta\frac{(N-i)i}{N},\]while a recovery rate may be
\[d(i)=\gamma i.\]5. Competing events
When several event types are possible, their rates compete. The total rate controls when the next event occurs, while relative rates determine which event occurs.