Inprobability 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]
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.)
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)
^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. S2CID14257280.
^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. JSTOR1427443. S2CID124108500.
^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. JSTOR3214906. S2CID122273001.
^Chennetier, G. (2022). "Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process". arXiv:2210.16185 [stat.CO].
