In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, 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
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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.
The model was first introduced in a paper by
Mark H. A. Davis
Mark Herbert Ainsworth Davis (25 April 1945 – 18 March 2020) was Professor of Mathematics at Imperial College London. He made fundamental contributions to the theory of stochastic processes, stochastic control and mathematical finance.
Edu ...
in 1984.
Examples
Piecewise linear models such as
Markov chains,
continuous-time Markov chain
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of ...
s, the
M/G/1 queue
In queueing theory, a discipline within the mathematical theory of probability, an M/G/1 queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there is a single server ...
, the
GI/G/1 queue In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times h ...
and the
fluid queue can be encapsulated as PDMPs with simple differential equations.
Applications
PDMPs have been shown useful in
ruin theory
In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the prob ...
,
queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
, for modelling
biochemical processes such as DNA replication in
eukaryotes and subtilin production by the organism
B. subtilis
''Bacillus subtilis'', known also as the hay bacillus or grass bacillus, is a Gram-positive, catalase-positive bacterium, found in soil and the gastrointestinal tract of ruminants, humans and marine sponges. As a member of the genus ''Bacillu ...
, and for modelling
earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, fr ...
s. Moreover, this class of processes has been shown to be appropriate for biophysical neuron models with stochastic ion channels.
Properties
Löpker and Palmowski have shown conditions under which a
time reversed PDMP is a PDMP. General conditions are known for PDMPs to be stable.
Galtier and Al. 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
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally at ...
, this work was further developed by Chennetier and Al.
to estimate the reliability of industrial systems.)
See also
*
Jump diffusion Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and pattern theory and computational vision.
In ph ...
, a generalization of piecewise-deterministic Markov processes
*
Hybrid system A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the te ...
(in the context of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s), a broad class of dynamical systems that includes all jump diffusions (and hence all piecewise-deterministic Markov processes)
References
Markov processes
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