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This book is about evaluating and optimizing the performance of continuous-time dynamical systems under uncertainty (more specifically, those systems in which the basic source of uncertainty is a sequence of random occurrences taking place at deterministic or random times). This covers an enormous variety of applications in engineering systems, operations research, management science, economics and applied probability a few examples are queueing systems (the random occurrences being arrival of customers or completion of service), investment planning (changes of interest rate or demand level), stochastic scheduling (completion of jobs or failure of machines), naval target tracking (changes of course by a manoeuvring target), insurance analysis (occurrence of claims, changes in premium rates), and optimal exploitation of resources such as fisheries, forestry or oil (amounts of resources found, random factors associated with harvesting or production, changes in market prices). All these examples- and there are many more- are dynamic in that actions are taken over time and actions taken now have repercussions in the future, and stochastic in that they involve uncertainty of the sort just described.
In this book a class of stochastic models called piecewise-deterministic Markov processes (referred to throughout as PDPs) is proposed as a general framework for studying problems of this kind. In the first half of the book (Chapters 1-3) the PDP is introduced, its properties are studied and methods are described for evaluating the probabilities of events and expectations and moments of random variables in stochastic system models. The second half of the book (Chapters 4 and 5) concerns optimization, i.e. stochastic control theory for PDP models. It differs radically, however, from other treatments of stochastic control, and the following remarks are intended to set the subject in context.
Analysis, probability and stochastic processes.
Piecewise-deterministic Markov processes.
Distributions and expectations.
Control theory.
Control by intervention.
Jump processes and their martingales