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]]."<ref name="davis" /> The process is defined by three quantities: the flow, the jump rate, and the transition measure.<ref name="siam2010">{{citeCite journal doi| last1 = Costa | first1 = O. L. V. | last2 = Dufour | first2 = F. | doi = 10.1137/080718541 | title = Average Continuous Control of Piecewise Deterministic Markov Processes | journal = SIAM Journal on Control and Optimization | volume = 48 | issue = 7 | pages = 4262 | year = 2010 | pmid = | pmc = | arxiv = 0809.0477}}</ref>
The model was first introduced in a paper by [[Mark H. A. Davis]] in 1984.<ref name="davis">{{cite jstor|2345677}}</ref>
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==Applications==
PDMPs have been shown useful in [[ruin theory]],<ref>{{cite jstor|1427443}}</ref> [[queueing theory]],<ref>{{cite jstor|3214906}}</ref><ref>{{citeCite journal doi| last1 = Boxma | first1 = O. | authorlink1 = Onno Boxma| last2 = Kaspi | first2 = H. | last3 = Kella | first3 = O. | last4 = Perry | first4 = D. | title = On/off Storage Systems with State-Dependent Input, Output, and Switching Rates | doi = 10.1017/S0269964805050011 | journal = Probability in the Engineering and Informational Sciences | volume = 19 | year = 2005 | pmid = | pmc = }}</ref> for modelling [[biochemistry|biochemical processes]] such as subtilin production by the organism [[B. subtilis]] and DNA replication in [[eukaryotes]]<ref>{{cite book|chapter=Chapter 9. Stochastic Hybrid Modeling of Biochemical Processes|chapterurl=http://www.nt.ntnu.no/users/skoge/prost/proceedings/hygea-workshop-july07-systems_biology/publications/JL2/Chapter9.pdf|title=Stochastic Hybrid Systems|first1=Christos G.|last1=Cassandras|first2=John|last2=Lygeros|publisher=CRC Press|year=2007|isbn=9780849390838}}</ref> for modelling [[earthquake]]s.<ref>{{citeCite journal doi| last1 = Ogata | first1 = Y. | last2 = Vere-Jones | first2 = D. | doi = 10.1016/0304-4149(84)90009-7 | title = Inference for earthquake models: A self-correcting model | journal = Stochastic Processes and their Applications | volume = 17 | issue = 2 | pages = 337 | year = 1984 | pmid = | pmc = }}</ref> Moreover, this class of processes has been shown to be appropriate for biophysical neuron models with stochastic ion channels.<ref>{{cite journal|last=Pakdaman|first=K.|author2=Thieullen, M. |author3=Wainrib, G. |title=Fluid limit theorems for stochastic hybrid systems with application to neuron models|journal=Advances in Applied Probability|date=September 2010|volume=42|issue=3|pages=761–794|doi=10.1239/aap/1282924062|url=https://sites.google.com/site/gwainrib/papers}}</ref>
==Properties==
Löpker and Palmowski have shown conditions under which a [[reversed process|time reversed]] PDMP is a PDMP.<ref>{{citeCite journal doi| last1 = Löpker | first1 = A. | last2 = Palmowski | first2 = Z. | doi = 10.1214/EJP.v18-1958 | title = On time reversal of piecewise deterministic Markov processes | journal = Electronic Journal of Probability | volume = 18 | year = 2013 | url = http://arxiv.org/pdf/1110.3813v1.pdf| pmid = | pmc = }}</ref> General conditions are known for PDMPs to be stable.<ref>{{citeCite journal doi| last1 = Costa | first1 = O. L. V. | last2 = Dufour | first2 = F. | doi = 10.1137/060670109 | title = Stability and Ergodicity of Piecewise Deterministic Markov Processes | journal = SIAM Journal on Control and Optimization | volume = 47 | issue = 2 | pages = 1053 | year = 2008 | pmid = | pmc = }}</ref>
