Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes E-Book

Benoite de Saporta

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 329

Veröffentlichungsjahr: 2015

Beschreibung

Mark H.A. Davis introduced the Piecewise-Deterministic Markov Process (PDMP) class of stochastic hybrid models in an article in 1984. Today it is used to model a variety of complex systems in the fields of engineering, economics, management sciences, biology, Internet traffic, networks and many more. Yet, despite this, there is very little in the way of literature devoted to the development of numerical methods for PDMDs to solve problems of practical importance, or the computational control of PDMPs.

This book therefore presents a collection of mathematical tools that have been recently developed to tackle such problems. It begins by doing so through examples in several application domains such as reliability. The second part is devoted to the study and simulation of expectations of functionals of PDMPs. Finally, the third part introduces the development of numerical techniques for optimal control problems such as stopping and impulse control problems.

Leseprobe

Table of Contents

Cover

Title

Copyright

Preface

Introduction

Part 1: Piecewise Deterministic Markov Processes and Quantization

1 Piecewise Deterministic Markov Processes

1.1. Introduction

1.2. Notation

1.3. Definition of a PDMP

1.4. Regularity assumptions

1.5. Time-augmented process

1.6. Embedded Markov chain

1.7. Stopping times

1.8. Examples of PDMPs

2 Examples in Reliability

2.1. Introduction

2.2. Structure subject to corrosion

2.3. The heated hold-up tank

3 Quantization Technique

3.1. Introduction

3.2. Optimal quantization

3.3. Simulation of PDMPs

3.4. Quantization of PDMPs

Part 2: Simulation of Functionals

4 Expectation of Functionals

4.1. Introduction

4.2. Recursive formulation

4.3. Lipschitz regularity

4.4. Rate of convergence

4.5. Time-dependent functionals

4.6. Deterministic time horizon

4.7. Example

4.8. Conclusion

5 Exit Time

5.1. Introduction

5.2. Problem setting

5.3. Approximation schemes

5.4. Convergence

5.5. Example

5.6. Conclusion

6 Example in Reliability: Service Time

6.1. Mean thickness loss

6.2. Service time

6.3. Conclusion

Part 3: Optimization

7 Optimal Stopping

7.1. Introduction

7.2. Dynamic programming equation

7.3. Approximation of the value function

7.4. Lipschitz continuity properties

7.5. Error estimation for the value function

7.6. Numerical construction of an e-optimal stopping time

7.7. Example

8 Partially Observed Optimal Stopping Problem

8.1. Introduction

8.2. Problem formulation and assumptions

8.3. Optimal filtering

8.4. Dynamic programming

8.5. Numerical approximation by quantization

8.6. Numerical example

9 Example in Reliability: Maintenance Optimization

9.1. Introduction

9.2. Corrosion process

9.3. Air conditioning unit

9.4. The heated hold-up tank

9.5. Conclusion

10 Optimal Impulse Control

10.1. Introduction

10.2. Impulse control problem

10.3. Lipschitz-continuity properties

10.4. Approximation of the value function

10.5. Example

10.6. Conclusion

Bibliography

Index

End User License Agreement

Guide

Cover

Table of Contents

Begin Reading

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Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes

Application to Reliability

Benoîte de Saporta

François Dufour

Huilong Zhang

Series EditorNikolaos Limnios

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2016

The rights of Benoîte de Saporta, François Dufour and Huilong Zhang to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2015954427

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-839-0