Markov models and optimization.

*(English)*Zbl 0780.60002
Monographs on Statistics and Applied Probability. 49. London: Chapman & Hall. xiv, 295 p. (1993).

The book deals with piecewise-deterministic processes (PDPs) which consist of a mixture of deterministic motion and random jumps. Although the structure of PDPs is very simple, they successfully describe numerous applied problems arising in engineering, operations research, management and economics.

In the first part of the book (Chapters 1-3) the PDPs are introduced and characterized as strong Markov and Borel right processes, their generators are investigated, the formula of change of variables is derived, and certain functionals of PDPs are characterized as solutions of integro-differential equations. In addition, relations between PDPs, stationary distributions and imbedded Markov chains are investigated. The second part of the book (Chapters 4 and 5) deals with continuous control problems (including relaxed controls) as well as impulse control ones for PDPs. The value functions are characterized as solutions of the integro- differential Bellman equations or quasi-variational inequalities.

In the first part of the book (Chapters 1-3) the PDPs are introduced and characterized as strong Markov and Borel right processes, their generators are investigated, the formula of change of variables is derived, and certain functionals of PDPs are characterized as solutions of integro-differential equations. In addition, relations between PDPs, stationary distributions and imbedded Markov chains are investigated. The second part of the book (Chapters 4 and 5) deals with continuous control problems (including relaxed controls) as well as impulse control ones for PDPs. The value functions are characterized as solutions of the integro- differential Bellman equations or quasi-variational inequalities.

Reviewer: H.Pragarauskas (Vilnius)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60J75 | Jump processes (MSC2010) |

93E20 | Optimal stochastic control |