science

Piecewise-deterministic Markov process

Piecewise-deterministic Markov process is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Piecewise-deterministic Markov process rather than just read about it. In short: In probability 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." The process is defined by three quan…

Key takeaways

Piecewise-deterministic Markov process belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Piecewise-deterministic Markov process to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Piecewise-deterministic Markov process from memory before moving on to harder problems.

Reference excerpt

In probability 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." 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 in 1984.

Examples Piecewise linear models such as Markov chains, continuous-time Markov chains, the M/G/1 queue, the GI/G/1 queue and the fluid queue can be encapsulated as PDMPs with simple differential equations.

Applications PDMPs have been shown useful in ruin theory, queueing theory, for modelling biochemical processes such as DNA replication in eukaryotes and subtilin production by the organism B. subtilis, and for modelling earthquakes. 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 et 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, this work was further developed by Chennetier and Al. to estimate the reliability of industrial systems.)

See also 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)

References

Worked examples

Example 1 — a first encounter with Piecewise-deterministic Markov process

Start with the simplest possible case. Write down what Piecewise-deterministic Markov process claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Piecewise-deterministic Markov process before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Piecewise-deterministic Markov process ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Piecewise-deterministic Markov process

In research: Piecewise-deterministic Markov process appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Piecewise-deterministic Markov process in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Piecewise-deterministic Markov process is common in secondary-school and first-year university syllabi. It links to neighbouring topics Markov processes, so understanding it makes those chapters shorter.
In everyday life: Look for Piecewise-deterministic Markov process outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Piecewise-deterministic Markov process in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Piecewise-deterministic Markov process means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Piecewise-deterministic Markov process out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Piecewise-deterministic Markov process in simple terms?

Why does Piecewise-deterministic Markov process matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Piecewise-deterministic Markov process?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Piecewise-deterministic Markov process.