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Piecewise-deterministic Markov process
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<h2 id="examples">Examples</h2> <p>Piecewise linear models such as <a href="/facts/Markov_chain/5oP5hntk">Markov chains</a>, <a href="/facts/Continuous-time_Markov_chain/h7ytJ6p3">continuous-time Markov chains</a>, the <a href="/facts/M%2fG%2f1_queue/bRA0YIHY">M/G/1 queue</a>, the <a href="/facts/GI%2fG%2f1_queue/pcfXFDIR">GI/G/1 queue</a> and the <a href="/facts/Fluid_queue/M6o7JMej">fluid queue</a> can be encapsulated as PDMPs with simple differential equations.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="applications">Applications</h2> <p>PDMPs have been shown useful in <a href="/facts/Ruin_theory/Jq3MaHIS">ruin theory</a>,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <a href="/facts/Queueing_theory/8NDl56EM">queueing theory</a>,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> for modelling <a href="/facts/Biochemistry/fa2pWEBp">biochemical processes</a> such as DNA replication in <a href="/facts/Eukaryotes/SkwyPLMA">eukaryotes</a> and subtilin production by the organism <a href="/facts/B._subtilis/KA8AodsE">B. subtilis</a>,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> and for modelling <a href="/facts/Earthquake/NGjIzagw">earthquakes</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> Moreover, this class of processes has been shown to be appropriate for biophysical neuron models with stochastic ion channels.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="properties">Properties</h2> <p>Löpker and Palmowski have shown conditions under which a <a href="/facts/Reversed_process/Jv4CFpSy">time reversed</a> PDMP is a PDMP.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> General conditions are known for PDMPs to be stable.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p><p>Galtier et al.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> 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 <a href="/facts/Importance_sampling/jrbUnMXa">importance sampling</a>, this work was further developed by Chennetier and Al.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> to estimate the reliability of industrial systems.) </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Jump_diffusion/Dw8ADE51">Jump diffusion</a>, a generalization of piecewise-deterministic Markov processes</li> <li><a href="/facts/Hybrid_system/wk7Dfdup">Hybrid system</a> (in the context of <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a>), a broad class of dynamical systems that includes all jump diffusions (and hence all piecewise-deterministic Markov processes)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Davis, M. H. A. (1984). "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. doi:10.1111/j.2517-6161.1984.tb01308.x. JSTOR 2345677. <a href="/wiki/Mark_H._A._Davis" target="_blank">/wiki/Mark_H._A._Davis</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Costa, O. L. V.; Dufour, F. (2010). "Average Continuous Control of Piecewise Deterministic Markov Processes". SIAM Journal on Control and Optimization. 48 (7): 4262. arXiv:0809.0477. doi:10.1137/080718541. S2CID 14257280. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Davis, M. H. A. (1984). "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. doi:10.1111/j.2517-6161.1984.tb01308.x. JSTOR 2345677. <a href="/wiki/Mark_H._A._Davis" target="_blank">/wiki/Mark_H._A._Davis</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Davis, M. H. A. (1984). "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. doi:10.1111/j.2517-6161.1984.tb01308.x. JSTOR 2345677. <a href="/wiki/Mark_H._A._Davis" target="_blank">/wiki/Mark_H._A._Davis</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Embrechts, P.; Schmidli, H. (1994). "Ruin Estimation for a General Insurance Risk Model". Advances in Applied Probability. 26 (2): 404–422. doi:10.2307/1427443. JSTOR 1427443. S2CID 124108500. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Browne, Sid; Sigman, Karl (1992). "Work-Modulated Queues with Applications to Storage Processes". Journal of Applied Probability. 29 (3): 699–712. doi:10.2307/3214906. JSTOR 3214906. S2CID 122273001. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Boxma, O.; Kaspi, H.; Kella, O.; Perry, D. (2005). "On/off Storage Systems with State-Dependent Input, Output, and Switching Rates". Probability in the Engineering and Informational Sciences. 19: 1–14. CiteSeerX 10.1.1.556.6718. doi:10.1017/S0269964805050011. S2CID 24065678. <a href="/wiki/Onno_Boxma" target="_blank">/wiki/Onno_Boxma</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Cassandras, Christos G.; Lygeros, John (2007). "Chapter 9. Stochastic Hybrid Modeling of Biochemical Processes" (PDF). Stochastic Hybrid Systems. CRC Press. ISBN 9780849390838. <a href="9780849390838" target="_blank">9780849390838</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Ogata, Y.; Vere-Jones, D. (1984). "Inference for earthquake models: A self-correcting model". Stochastic Processes and Their Applications. 17 (2): 337. doi:10.1016/0304-4149(84)90009-7. <a href="https://doi.org/10.1016%2F0304-4149%2884%2990009-7" target="_blank">https://doi.org/10.1016%2F0304-4149%2884%2990009-7</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Pakdaman, K.; Thieullen, M.; Wainrib, G. (September 2010). "Fluid limit theorems for stochastic hybrid systems with application to neuron models". Advances in Applied Probability. 42 (3): 761–794. arXiv:1001.2474. doi:10.1239/aap/1282924062. S2CID 18894661. <a href="https://sites.google.com/site/gwainrib/papers" target="_blank">https://sites.google.com/site/gwainrib/papers</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Löpker, A.; Palmowski, Z. (2013). "On time reversal of piecewise deterministic Markov processes". Electronic Journal of Probability. 18. arXiv:1110.3813. doi:10.1214/EJP.v18-1958. S2CID 1453859. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Costa, O. L. V.; Dufour, F. (2008). "Stability and Ergodicity of Piecewise Deterministic Markov Processes" (PDF). SIAM Journal on Control and Optimization. 47 (2): 1053. doi:10.1137/060670109. <a href="http://www.producao.usp.br/bitstream/BDPI/14708/1/art_COSTA_Stability_and_ergodicity_of_piecewise_deterministic_Markov_2008.pdf" target="_blank">http://www.producao.usp.br/bitstream/BDPI/14708/1/art_COSTA_Stability_and_ergodicity_of_piecewise_deterministic_Markov_2008.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Galtier, T. (2019). "On the optimal importance process for piecewise deterministic Markov process". Esaim: Ps. 23: 893–921. doi:10.1051/ps/2019015. S2CID 198467101. <a href="https://doi.org/10.1051%2Fps%2F2019015" target="_blank">https://doi.org/10.1051%2Fps%2F2019015</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Chennetier, G. (2022). "Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process". arXiv:2210.16185 [stat.CO]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> </ol>