Diffusions, Markov Processes and Martingales

Foundations v. 1

By: L. C. G. Rogers, David Williams

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Paperback | 13 April 2000 | Edition Number 2

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406 Pages

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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.

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Non-Fiction Mathematics Probability & Statistics Applied Mathematics Stochastics Number Theory

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

Diffusions, Markov Processes and Martingales

Foundations v. 1

At a Glance

Paperback

Shipping

How to return your order

Defective items

You Can Find This Book In

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