Piecewise deterministic Markov process — recent results

Abstract

Abstract.We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the inter-jumping times. We conclude with a short overview on numerical methods used for simulating PDMPs. General introductionThe piecewise deterministic Markov processes (denoted PDMPs) were first introduced in the literature by Davis ( [Dav84,Dav93]). Already at this time, the theory of diffusions had such powerful tools as the theory of Itō calculus and stochastic differential equations at its disposal. Davis's goal was to endow the PDMP with rather general tools. The main reason for that was to provide a general framework, since up to then only very particular cases had been dealt with, which turned out not to be easily generalizable.PDMPs form a family of càdlàg Markov processes involving a deterministic motion punctuated by random jumps. The motion of the PDMP {X(t)} t≥0 depends on three local characteristics, namely the jump rate λ, the flow φ and the transition measure Q according to which the location of the process at the jump time is chosen. The process starts from x and follows the flow φ(x, t) until the first jump time T 1 which occurs either spontaneously in a Poisson-like fashion with rate λ (φ(x, t)) or when the flow φ(x, t) hits the boundary of the state-space. In both cases, the location of the process at the jump time T 1 , denoted by Z 1 = X(T 1 ), is selected by the transition measure Q(φ(x, T 1 ), ·) and the motion restarts from this new point as before. This fully describes a piecewise continuous trajectory for {X(t)} with jump times {T k } and post jump locations {Z k }, and which evolves according to the flow φ between two jumps. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

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