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Piecewise-deterministic Markov processes: A general class of non- diffusion stochastic models. (English) Zbl 0565.60070
A class of Markov processes whose trajectories are of deterministic motions with random jumps is under consideration. The author develops stochastic calculus for such processes. He proves an extension of Dynkin’s formula, gives a complete characterization of the extended generator and receives a ”Feynman-Kac” formula. Some control problems for these processes are described. The author uses martingales when proving his results.
The paper contains also a discussion of the above results with contributions by well-known mathematicians (D.R. Cox, M. S. Bartlett, D. P. Kennedy, F. P. Kelly and many others).
Reviewer’s remark: The above processes were studied in the following books - the reviewer, Qualitative analysis of the behaviour of complex systems by the method of test functions. (1978; Zbl 0451.93002); N. P. Buslenko, the reviewer and I. N. Kovalenko, Lectures on complex systems theory. (1973; Zbl 0252.93001), see also the reviewer, Teor. Veroyatn. Primen. 20, 571-583 (1975; Zbl 0361.60034); English translation in Theory Probab. Appl. 20, 560-571 (1975).
In particular, the first book contains the extension of Dynkin’s formula, characterization of the extended generator, some results concerning distributions of first passage times (see D. Vermes’ remark in the discussion), boundedness, regularity and other properties. The author mentioned these overlappings and used other approaches (e.g. martingales) and obtained further results.
Reviewer: V.V.Kalashnikov

60J99 Markov processes
60G44 Martingales with continuous parameter
60H99 Stochastic analysis
60K25 Queueing theory (aspects of probability theory)