Diffusions, Markov Processes, and Martingales

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ISBN: 9780521775946 | 0521775949
Cover: Paperback
Copyright: 5/1/2000

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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.

Some Frequently Used Notation

xix

Brownian Motion

Introduction

(9)

What is Brownian motion, and why study it?

(1)

Brownian motion as a martingale

(1)

Brownian motion as a Gaussian process

(2)

Brownian motion as a Markov process

(2)

Brownian motion as a diffusion (and martingale)

(3)

Basics About Brownian Motion

(26)

Existence and uniqueness of Brownian motion

(3)

Skorokhod embedding

(3)

Donsker's Invariance Principle

(2)

Exponential martingales and first-passage distributions

(1)

Some sample-path properties

(2)

Quadratic variation

(1)

The strong Markov property

(4)

Reflection

(2)

Reflecting Brownian motion and local time

(4)

Kolmogorov's test

(1)

Brownian exponential martingales and the Law of the Iterated Logarithm

(5)

Brownian Motion in Higher Dimensions

(19)

Some martingales for Brownian motion

(2)

Recurrence and transience in higher dimensions

(1)

Some applications of Brownian motion to complex analysis

(4)

Windings of planar Brownian motion

(2)

Multiple points, cone points, cut points

(1)

Potential theory of Brownian motion in IRd (d≥3)

(5)

Brownian motion and physical diffusion

(4)

Gaussian Processes and Levy Processes

(30)

Gaussian processes

Existence results for Gaussian processes

(4)

Continuity results

(7)

Isotropic random flows

(5)

Dynkin's Isomorphism Theorem

(2)

Levy processes

(7)

Fluctuation theory and Wiener--Hopf factorisation

(2)

Local time of Levy processes

(3)

Some Classical Theory

Basic Measure Theory

(23)

Measurability and measure

Measurable spaces; σ-algebras; π-systems; d-systems

(3)

Measurable functions

(2)

Monotone-Class Theorems

(1)

Measures; the uniqueness lemma; almost everywhere; a.e.(μ,Σ)

(2)

Caratheodory's Extension Theorem

(1)

Inner and outer μ-measures; completion

(1)

Integration

Definition of the integral ∫ f dμ

(1)

Convergence theorems

(2)

The Radon-Nikodym Theorem; absolute continuity; λ « μ notation; equivalent measures

(1)

Inequalities; and Lp spaces (p≥1)

(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

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Diffusions, Markov Processes, and Martingales

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Table of Contents

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