Piecewise-deterministic Markov process

Add links
Source: Wikipedia, the free encyclopedia.
"PDMP" redirects here. For prescription drug monitoring programs, see Prescription monitoring program.

In probability theory, a piecewise-deterministic Markov process (PDMP) is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an ordinary differential equation between those times. The class of models is "wide enough to include as special cases virtually all the non-diffusion models of applied probability."[1] The process is defined by three quantities: the flow, the jump rate, and the transition measure.[2]

The model was first introduced in a paper by Mark H. A. Davis in 1984.[1]

Examples

Piecewise linear models such as

GI/G/1 queue and the fluid queue can be encapsulated as PDMPs with simple differential equations.[1]

Applications

PDMPs have been shown useful in

B. subtilis,[6] and for modelling earthquakes.[7] Moreover, this class of processes has been shown to be appropriate for biophysical neuron models with stochastic ion channels.[8]

Properties

Löpker and Palmowski have shown conditions under which a

time reversed PDMP is a PDMP.[9] General conditions are known for PDMPs to be stable.[10]

Galtier and Al.[11] studied the law of the trajectories of PDMP and provided a reference measure in order to express a density of a trajectory of the PDMP. Their work opens the way to any application using densities of trajectory. (For instance, they used the density of a trajectories to perform importance sampling, this work was further developed by Chennetier and Al.[12] to estimate the reliability of industrial systems.)

See also

  • Jump diffusion, a generalization of piecewise-deterministic Markov processes
  • Hybrid system (in the context of dynamical systems), a broad class of dynamical systems that includes all jump diffusions (and hence all piecewise-deterministic Markov processes)

References

  1. ^
    JSTOR 2345677
    .
  2. S2CID 14257280
    .
  3. S2CID 124108500
    .
  4. S2CID 122273001
    .
  5. S2CID 24065678
    .
  6. ISBN 9780849390838
    .
  7. doi:10.1016/0304-4149(84)90009-7
    .
  8. S2CID 18894661
    .
  9. S2CID 1453859
    .
  10. doi:10.1137/060670109
    .
  11. S2CID 198467101
    .
  12. arXiv:2210.16185 [stat.CO
    ].


Stub icon

This probability-related article is a stub. You can help Wikipedia by expanding it.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Piecewise-deterministic_Markov_process&oldid=1171977140"