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 o ...
, 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 w ...
between those times. The class of models is "wide enough to include as special cases virtually all the non-diffusion models of
applied probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains.
Scope
Much research involving probability is done under the auspices of applied probability. However, while such res ...
."
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 chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
s,
continuous-time Markov chains, the
M/G/1 queue, 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 In queueing theory, a discipline within the mathematical theory of probability, a fluid queue (fluid model, fluid flow model or stochastic fluid model) is a mathematical model used to describe the fluid level in a reservoir subject to randomly deter ...
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, for modelling
biochemical processes
Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology an ...
such as DNA replication in
eukaryotes
Eukaryotes () are organisms whose cells have a nucleus. All animals, plants, fungi, and many unicellular organisms, are Eukaryotes. They belong to the group of organisms Eukaryota or Eukarya, which is one of the three domains of life. Bacte ...
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 ''Bacillus' ...
, 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, from ...
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 att ...
, 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 p ...
, a generalization of piecewise-deterministic Markov processes
*
Hybrid system (in the context of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s), a broad class of dynamical systems that includes all jump diffusions (and hence all piecewise-deterministic Markov processes)
References
Markov processes
{{probability-stub