Diffusions, Markov Processes, and Martingales

by L. C. G. Rogers , David Williams

• ISBN13: 9780521775946
• ISBN10: 0521775949
• Edition: 2nd
• Format: Paperback
• Copyright: 2000-05-01
• Publisher: Cambridge University Press

Summary

Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Table of Contents

Some Frequently Used Notation	xix
Brownian Motion
Introduction	1	(9)
What is Brownian motion, and why study it?	1	(1)
Brownian motion as a martingale	2	(1)
Brownian motion as a Gaussian process	3	(2)
Brownian motion as a Markov process	5	(2)
Brownian motion as a diffusion (and martingale)	7	(3)
Basics About Brownian Motion	10	(26)
Existence and uniqueness of Brownian motion	10	(3)
Skorokhod embedding	13	(3)
Donsker's Invariance Principle	16	(2)
Exponential martingales and first-passage distributions	18	(1)
Some sample-path properties	19	(2)
Quadratic variation	21	(1)
The strong Markov property	21	(4)
Reflection	25	(2)
Reflecting Brownian motion and local time	27	(4)
Kolmogorov's test	31	(1)
Brownian exponential martingales and the Law of the Iterated Logarithm	31	(5)
Brownian Motion in Higher Dimensions	36	(19)
Some martingales for Brownian motion	36	(2)
Recurrence and transience in higher dimensions	38	(1)
Some applications of Brownian motion to complex analysis	39	(4)
Windings of planar Brownian motion	43	(2)
Multiple points, cone points, cut points	45	(1)
Potential theory of Brownian motion in IRd (d≥3)	46	(5)
Brownian motion and physical diffusion	51	(4)
Gaussian Processes and Levy Processes	55	(30)
Gaussian processes
Existence results for Gaussian processes	55	(4)
Continuity results	59	(7)
Isotropic random flows	66	(5)
Dynkin's Isomorphism Theorem	71	(2)
Levy processes
Levy processes	73	(7)
Fluctuation theory and Wiener--Hopf factorisation	80	(2)
Local time of Levy processes	82	(3)
Some Classical Theory
Basic Measure Theory	85	(23)
Measurability and measure
Measurable spaces; σ-algebras; π-systems; d-systems	85	(3)
Measurable functions	88	(2)
Monotone-Class Theorems	90	(1)
Measures; the uniqueness lemma; almost everywhere; a.e.(μ,Σ)	91	(2)
Caratheodory's Extension Theorem	93	(1)
Inner and outer μ-measures; completion	94	(1)
Integration
Definition of the integral ∫ f dμ	95	(1)
Convergence theorems	96	(2)
The Radon-Nikodym Theorem; absolute continuity; λ « μ notation; equivalent measures	98	(1)
Inequalities; and Lp spaces (p≥1)	99	(2)
Product structures
Product σ-algebras	101	(1)
Product measure; Fubini's Theorem	102	(2)
Exercises	104	(4)
Basic Probability Theory	108	(11)
Probability and expectation
Probability triple; almost surely (a.s.); a.s.(P), a.s.(P,F)	108	(1)
lim sup En; First Borel--Cantelli Lemma	109	(1)
Law of random variable; distribution function; joint law	110	(1)
Expectation; E(X;F)	110	(1)
Inequalities: Markov, Jensen, Schwarz, Tchebychev	111	(2)
Modes of convergence of random variables	113	(1)
Uniform integrability and 1 convergence
Uniform integrability	114	(1)
1 convergence	115	(1)
Independence
Independence of σ-algebras and of random variables	116	(2)
Existence of families of independent variables	118	(1)
Exercises	119	(1)
Stochastic Processes	119	(18)
The Daniell--Kolmogorov Theorem
(ET,ET); σ-algebras on function space; cylinders and σ-cylinders	119	(2)
Infinite products of probability triples	121	(1)
Stochastic process; sample function; law	121	(1)
Canonical process	122	(1)
Finite-dimensional distributions; sufficiency; compatibility	123	(1)
The Daniell--Kolmogorov (DK) Theorem: `compact metrizable' case	124	(2)
The Daniell--Kolmogorov (DK) Theorem: general case	126	(1)
Gaussian processes; pre-Brownian motion	127	(1)
Pre-Poisson set functions	128	(1)
Beyond the DK Theorem
Limitations of the DK Theorem	128	(1)
The role of outer measures	129	(1)
Modifications; indistinguishability	130	(1)
Direct construction of Poisson measures and subordinators, and of local time from the zero set; Azema's martingale	131	(5)
Exercises	136	(1)
Discrete-Parameter Martingale Theory	137	(26)
Conditional expectation
Fundamental theorem and definition	137	(1)
Notation; agreement with elementary usage	138	(1)
Properties of conditional expectation: a list	139	(1)
The role of versions; regular conditional probabilities and pdfs	140	(1)
A counterexample	141	(1)
A uniform-integrability property of conditional expectations	142	(1)
(Discrete-parameter) martingales and supermartingales
Filtration; filtered space; adapted process; natural filtration	143	(1)
Martingale; supermartingale; submartingale	144	(1)
Previsible process; gambling strategy; a fundamental principle	144	(1)
Doob's Upcrossing Lemma	145	(1)
Doob's Supermartingale-Convergence Theorem	146	(1)
1 convergence and the UI property	147	(1)
The Levy--Doob Downward Theorem	148	(2)
Doob's Submartingale and Inequalities	150	(2)
Martingales in 2; orthogonality of increments	152	(1)
Doob decomposition	153	(1)
The <M> and [M] processes	154	(1)
Stopping times, optional stopping and optional sampling
Stopping time	155	(1)
Optional-stopping theorems	156	(2)
The pre-T σ-algebra FT	158	(1)
Optional sampling	159	(2)
Exercises	161	(2)
Continuous-Parameter Supermartingales	163	(37)
Regularisation: R-supermartingales
Orientation	163	(1)
Some real-variable results	163	(3)
Filtrations; supermartingales; R-processes, R-supermartingales	166	(1)
Some important examples	167	(2)
Doob's Regularity Theorem: Part 1	169	(2)
Partial augmentation	171	(1)
Usual conditions; R-filtered space; usual augmentation; R-regularisation	172	(2)
A necessary pause for thought	174	(1)
Convergence theorems for R-supermartingales	175	(2)
Inequalities and convergence for R-submartingales	177	(1)
Martingale proof of Wiener's Theorem; canonical Brownian motion	178	(2)
Brownian motion relative to a filtered space	180	(1)
Stopping times
Stopping time T; pre-T σ-algebra GT; progressive process	181	(2)
First-entrance (debut) times; hitting times; first-approach times: the easy cases	183	(1)
Why `completion' in the usual conditions has to be introduced	184	(2)
Debut and Section Theorems	186	(2)
Optional Sampling for R-supermartingales under the usual conditions	188	(3)
Two important results for Markov-process theory	191	(1)
Exercises	192	(8)
Probability Measures on Lusin Spaces	200	(27)
`Weak convergence'
C(J) and Pr(J) when J is compact Hausdorff	202	(1)
C(J) and Pr(J) when J is compact metrizable	203	(2)
Polish and Lusin spaces	205	(2)
The Cb(S) topology of Pr(S) when S is a Lusin space; Prohorov's Theorem	207	(4)
Some useful convergence results	211	(2)
Tightness in Pr(W) when W is the path-space W:=C([0,∞);IR)	213	(2)
The Skorokhod representation of Cb(S) convergence on Pr(S)	215	(1)
Weak convergence versus convergence of finite-dimensional distributions	216	(1)
Regular conditional probabilities
Some preliminaries	217	(1)
The main existence theorem	218	(2)
Canonical Brownian Motion CBM(IRN); Markov property of Px laws	220	(2)
Exercises	222	(5)
Markov Processes
Transition Functions and Resolvents	227	(13)
What is a (continuous-time) Markov process?	227	(1)
The finite-state-space Markov chain	228	(3)
Transition functions and their resolvents	231	(3)
Contraction semigroups on Banach spaces	234	(3)
The Hille--Yosida Theorem	237	(3)
Feller--Dynkin Processes	240	(23)
Feller--Dynkin (FD) semigroups	240	(3)
The existence theorem: canonical FD processes	243	(4)
Strong Markov property: preliminary version	247	(2)
Strong Markov property: full version; Blumenthal's 0--1 Law	249	(3)
Some fundamental martingales; Dynkin's formula	252	(3)
Quasi-left-continuity	255	(1)
Characteristic operator	256	(2)
Feller--Dynkin diffusions	258	(3)
Characterisation of continuous real Levy processes	261	(1)
Consolidation	262	(1)
Additive Functionals	263	(21)
PCHAFs; λ-excessive functions; Brownian local time	263	(4)
Proof of the Volkonskii--Sur--Meyer Theorem	267	(2)
Killing	269	(3)
The Feynmann--Kac formula	272	(3)
A Ciesielski--Taylor Theorem	275	(2)
Time-substitution	277	(1)
Reflecting Brownian motion	278	(3)
The Feller--McKean chain	281	(1)
Elastic Brownian motion; the arcsine law	282	(2)
Approach to Ray Processes: The Martin Boundary	284	(19)
Ray processes and Markov chains	284	(2)
Important example: birth process	286	(2)
Excessive functions, the Martin kernel and Choquet theory	288	(4)
The Martin compactification	292	(3)
The Martin representation; Doob--Hunt explanation	295	(2)
R. S. Martin's boundary	297	(1)
Doob--Hunt theory for Brownian motion	298	(4)
Ray processes and right processes	302	(1)
Ray Processes	303	(18)
Orientation	303	(1)
Ray resolvents	304	(2)
The Ray--Knight compactification	306	(3)
Ray's Theorem: analytical part
From semigroup to resolvent	309	(4)
Branch-points	313	(2)
Choquet representation of 1-excessive probability measures	315	(1)
Ray's Theorem: probabilistic part
The Ray process associated with a given entrance law	316	(2)
Strong Markov property of Ray processes	318	(1)
The role of branch-points	319	(2)
Applications	321	(30)
Martin boundary theory in retrospect
From discrete to continuous time	321	(2)
Proof of the Doob--Hunt Convergence Theorem	323	(2)
The Choquet representation of II-excessive functions	325	(2)
Doob h-transforms	327	(1)
Time reversal and related topics
Nagasawa's formula for chains	328	(2)
Strong Markov property under time reversal	330	(1)
Equilibrium charge	331	(1)
BM (IR) and BES(3): splitting times	332	(2)
A first look at Markov-chain theory
Chains as Ray processes	334	(3)
Significance of qi	337	(1)
Taboo probabilities; first-entrance decomposition	337	(2)
The Q-matrix; DK conditions	339	(1)
Local-character condition for Q	340	(2)
Totally instantaneous Q-matrices	342	(1)
Last exits	343	(2)
Excursions from b	345	(2)
Kingman's solution of the `Markov characterization problem'	347	(1)
Symmetrisable chains	348	(1)
An open problem	349	(2)
References for Volumes 1 and 2	351	(24)
Index to Volumes 1 and 2	375