Jump chain
Topic of Continuous-time Markov chains, part of Stochastic processes
Jump chain
| Type | Topic |
| Taught in | Continuous-time Markov chains |
| Part | Stochastic processes |
The discrete-time chain of states that a continuous-time Markov chain visits, ignoring how long it stays in each. From state i it moves to j\ne i with probability q_{ij}/q_i, where q_i=-q_{ii} is the total rate of leaving i.
In Continuous-time Markov chains
Main article: Holding times, the jump chain, and the generator
In state i the holding time is exponential with rate \nu_i. If \nu_i>0, the next state is j with probability k_{ij}=q_{ij}/\nu_i. The matrix K=(k_{ij}) is the embedded jump chain, a discrete-time transition matrix. An absorbing state has \nu_i=0: the chain never leaves.
Used in
See also
Generator · Kolmogorov forward and backward equations · Two-state continuous-time chain · Uniformisation · Holding time