Piecewise-deterministic Markov process explained

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." The process is defined by three quantities: the flow, the jump rate, and the transition measure.^[1]

The model was first introduced in a paper by Mark H. A. Davis in 1984.^[2]

Examples

Piecewise linear models such as Markov chains, continuous-time Markov chains, the M/G/1 queue, the GI/G/1 queue and the fluid queue can be encapsulated as PDMPs with simple differential equations.

Applications

PDMPs have been shown useful in ruin theory,^[3] queueing theory,^{[4] [5]} for modelling biochemical processes such as DNA replication in eukaryotes and subtilin production by the organism 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)

Notes and References

1. Costa . O. L. V. . Dufour . F. . 10.1137/080718541 . Average Continuous Control of Piecewise Deterministic Markov Processes . SIAM Journal on Control and Optimization . 48 . 7 . 4262 . 2010 . 0809.0477. 14257280 .
2. Davis . M. H. A. . Mark H. A. Davis. Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models . Journal of the Royal Statistical Society. Series B (Methodological). 46 . 3 . 353–388 . 10.1111/j.2517-6161.1984.tb01308.x. 2345677. 1984 .
3. Embrechts . P. . Schmidli . H. . Ruin Estimation for a General Insurance Risk Model . Advances in Applied Probability . 26 . 2 . 404–422 . 10.2307/1427443 . 1427443. 1994 . 124108500 .
4. Browne . Sid . Sigman . Karl . 1992 . Work-Modulated Queues with Applications to Storage Processes . Journal of Applied Probability . 29 . 3 . 699–712 . 3214906 . 10.2307/3214906 . 122273001 .
5. Boxma . O. . Onno Boxma. Kaspi . H. . Haya Kaspi. Kella . O. . Perry . D. . On/off Storage Systems with State-Dependent Input, Output, and Switching Rates . 10.1017/S0269964805050011 . Probability in the Engineering and Informational Sciences . 19 . 2005 . 1–14 . 10.1.1.556.6718 . 24065678 .
6. Book: Chapter 9. Stochastic Hybrid Modeling of Biochemical Processes. http://www.nt.ntnu.no/users/skoge/prost/proceedings/hygea-workshop-july07-systems_biology/publications/JL2/Chapter9.pdf. Stochastic Hybrid Systems. Christos G.. Cassandras. John. Lygeros. CRC Press. 2007. 9780849390838.
7. Ogata . Y. . Vere-Jones . D. . 10.1016/0304-4149(84)90009-7 . Inference for earthquake models: A self-correcting model . Stochastic Processes and Their Applications . 17 . 2 . 337 . 1984 . free .
8. Pakdaman. K.. Thieullen, M. . Wainrib, G. . Fluid limit theorems for stochastic hybrid systems with application to neuron models. Advances in Applied Probability. September 2010. 42. 3. 761–794. 10.1239/aap/1282924062. 1001.2474. 18894661 .
9. Löpker . A. . Palmowski . Z. . 10.1214/EJP.v18-1958 . On time reversal of piecewise deterministic Markov processes . Electronic Journal of Probability . 18 . 2013 . 1110.3813. 1453859 .
10. Costa . O. L. V. . Dufour . F. . 10.1137/060670109 . Stability and Ergodicity of Piecewise Deterministic Markov Processes . SIAM Journal on Control and Optimization . 47 . 2 . 1053 . 2008 .
11. Galtier . T. . 10.1051/ps/2019015 . On the optimal importance process for piecewise deterministic Markov process . Esaim: Ps . 23 . 2019 . 893–921 . 198467101 . free .
12. Chennetier . G. . Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process . 2022 . stat.CO . 2210.16185 .