Exponential Families of Stochastic Processes

by Kuchler, Uwe; Srensen, Michael

• ISBN13: 9780387949819
• ISBN10: 038794981X
• Format: Hardcover
• Copyright: 1997-09-01
• Publisher: Springer Verlag

Summary

This is author-approved bcc: This book provides a comprehensive account of the statistical theory of exponential families of stochastic processes. The book reviews the progress in the field made over the last ten years or so by the authors, two of the leading experts in the field, and several other researchers. The theory is applied to a broad spectrum of examples. A large number of frequently applied stochastic process models with discrete as well as with continuous time are covered by the theory developed in the book. The exponential families of stochastic processes are the most tractable type of statistical models for stochastic processes. On the other hand, they include models that are complex enough to exhibit basic inference problems that are peculiar to stochastic process models. Therefore they are a good starting point for the statistican who plans to work in this interesting and vigorous field. To make the reading easier for statisticians with only a basic background in the theory of stochastic process, the first part of the book is based on classical theory of stochastic processes only, while stochastic calculus is used late in the book. Most of the concepts and tools from stochastic calculus that a statistician is likely to need, when working with inference for stochastic processes, are introduced and explained without proof in an appendix. The appendix can also be used independently as an introduction to stochastic calculus for statisticians. The statistical concepts are explained carefully so that probabilists with only a basic background in statistics can use the book to get into statistical inference for stochastic processes. Exercises are included to make the book useful for an advanced

Table of Contents

Preface	v
1 Introduction	1	(6)
2 Natural Exponential Families of Levy Processes	7	(12)
2.1 Definition and probabilistic properties	7	(7)
2.2 Maximum likelihood estimation	14	(1)
2.3 Exercises	15	(1)
2.4 Bibliographic notes	16	(3)
3 Definitions and Examples	19	(18)
3.1 Basic definitions	19	(3)
3.2 Markov processes in discrete time	22	(1)
3.3 More general discrete time models	23	(1)
3.4 Counting processes and marked counting processes	24	(3)
3.5 Diffusion-type processes	27	(2)
3.6 Diffusion processes with jumps	29	(1)
3.7 Random fields	30	(1)
3.8 The significance of the filtration	31	(2)
3.9 Exercises	33	(1)
3.10 Bibliographic notes	34	(3)
4 First Properties	37	(8)
4.1 Exponential representations	37	(2)
4.2 Exponential families of stochastic processes with a non-empty kernel	39	(3)
4.3 Exercises	42	(1)
4.4 Bibliographic notes	43	(2)
5 Random Time Transformations	45	(20)
5.1 An important type of continuous time model	45	(3)
5.2 Statistical results	48	(6)
5.3 More general models	54	(1)
5.4 Inverse families	55	(2)
5.5 Discrete time models	57	(4)
5.6 Exercises	61	(1)
5.7 Bibliographic notes	62	(3)
6 Exponential Families of Markov Processes	65	(16)
6.1 Conditional exponential families	65	(2)
6.2 Markov processes	67	(4)
6.3 The structure of exponential families of Markov processes	71	(6)
6.4 Exercises	77	(1)
6.5 Bibliographic notes	78	(3)
7 The Envelope Families	81	(22)
7.1 General theory	81	(3)
7.2 Markov processes	84	(3)
7.3 Explicit calculations	87	(3)
7.4 The Gaussian autoregression	90	(4)
7.5 The pure birth process	94	(2)
7.6 The Ornstein-Uhlenbeck process	96	(2)
7.7 A goodness-of-fit test	98	(2)
7.8 Exercises	100	(1)
7.9 Bibliographic notes	101	(2)
8 Likelihood Theory	103	(32)
8.1 Likelihood martingales	103	(5)
8.2 Existence and uniqueness of the maximum likelihood estimator	108	(5)
8.3 Consistency and asymptotic normality of the maximum likelihood estimator	113	(11)
8.4 Information matrices	124	(4)
8.5 Local asymptotic mixed normality	128	(2)
8.6 Exercises	130	(3)
8.7 Bibliographic notes	133	(2)
9 Linear Stochastic Differential Equations with Time Delay	135	(22)
9.1 The differential equations and the maximum likelihood estimator	135	(4)
9.2 The fundamental solution x0(.)	139	(2)
9.3 Asymptotic likelihood theory	141	(4)
9.4 The case N = 1	145	(10)
9.5 Exercises	155	(1)
9.6 Bibliographic notes	156	(1)
10 Sequential Methods	157	(48)
10.1 Preliminary result	157	(4)
10.2 Exact likelihood theory	161	(12)
10.3 Asymptotic likelihood theory	173	(8)
10.4 Comparison of sampling times	181	(6)
10.5 Moments of the stopping times	187	(5)
10.6 The sequential probability ratio test	192	(9)
10.7 Exercises	201	(2)
10.8 Bibliographic notes	203	(2)
11 The Semimartingale Approach	205	(36)
11.1 The local characteristics of a semimartingale	205	(3)
11.2 The natural exponential family generated by a semimartingale	208	(8)
11.3 Exponential families with a time-continuous likelihood function	216	(5)
11.4 Other types of exponential families of semimartingales	221	(4)
11.5 General exponential families of Levy processes	225	(4)
11.6 Exponential families constructed by stochastic time transformation	229	(2)
11.7 Likelihood theory	231	(6)
11.8 Exercises	237	(1)
11.9 Bibliographic notes	238	(3)
12 Alternative Definitions	241	(26)
12.1 Exponential marginal distributions	242	(7)
12.2 Families with a sufficient reduction	249	(13)
12.3 Exercises	262	(3)
12.4 Bibliographic Notes	265	(2)
A A Toolbox from Stochastic Calculus	267	(32)
A.1 Stochastic basis	267	(2)
A.2 Local martingales and increasing processes	269	(2)
A.3 Doob-Meyer decomposition	271	(3)
A.4 Semimartingales and stochastic integration	274	(5)
A.5 Stochastic differential equations	279	(2)
A.6 Ito's formula	281	(3)
A.7 Martingale limit theorems	284	(4)
A.8 Stochastic integration with respect to random measures	288	(3)
A.9 Local characteristics of a semimartingale	291	(3)
A.10 A Girsanov-type theorem for semimartingales	294	(5)
B Miscellaneous Results	299	(4)
B.1 The fundamental identity of sequential analysis	299	(2)
B.2 A conditional Radon-Nikodym derivative	301	(1)
B.3 Three lemmas	301	(2)
C References	303	(14)
D Basic Notation	317	(2)
Index	319