Preface Suggestions for the Reader Interdependence of the Chapters Frequently Used Notation CHAPTER 1 Martingales，Stopping Times，and Filtrations 1.1.Stochastic Processes and d.Fields 1.2.Stopping Times 1.3.Continuous.Time Martingales A.Fundamental inequalities B.Convergence results C.The optional sampling theorem 1.4.The Doob—Meyer Decomposition l.5.Continuous，Square-Integrable Martingales 1.6.Solutions to Selected Problems 1.7.Notes CHAPTER 2 Brownian Motion 2.1.IntrOductiOn 2.2.First Construction of Brownian Motion A.The consistency theorem B.The Kolmogorov-Centsov theorem 2.3.Second Construction of Brownian Motion 2.4.The Space C[0，∞)，Weak Convergence，and Wiener Measure A.Weak convergenee B.Tightness C.Convergence of finite.dimensional distributions D.The invariance principle and the Wiener measure 2.5.The Markov Property A.Brownian motion in several dimensions B.MarkOV processes and Markov families C.Equivalent formulations of the Markov property 2.6.The Strong Markov Property and the Reflection Principle A.The reflection principie B.Strong Markov processes and families C.The strong Markov property for Brownian motion 2.7.Brownian Filtrations A.Right.continuity of the augmented filtration for a strong Markov process B.A「universal」filtration C.The Blumenthal zero.one law 2.8.Computations Based on Passage Times A.Brownian motion and its running maximum B.Brownian motion on a half-line C.Brownian motion on a finite interval D.Distributions involving last exit times 2.9.The Brownian Sample Paths A.Elementary properties B.The zero set and the quadratic variation C.Local maxima and points of increase D.Nowhere ditierentiability
E.Law of the iterated logarithm F.Modulus of continuity 2.10.Solutions to Selected Problems 2.11.Notes CHAPTER 3 Stochastic Integration 3.1.IntroductiOn 3.2.Construction of the Stochastic Integral A.Simple processes and approximations B.Construction and elementary properties of the integral C.A characterization of the integral D.Integration with respect to continuous，local martingales 3.3.The Change-of-Variable Formula A.The Ito rule B.Martingale characterization of Brownian motion C.Bessel processes.questions of recurrence D.Martingale moment inequalities E.Supplementary exercises 3.4.Representations of Continuous Martingales in Terms of Brownian Motion A.Continuous local martingales as stochastic integrals with respect to Brownian motion B.Continuous local martingales as time-changed Brownian motions C.A theorem of F.B.Knight D.Brownian martingales as stochastic integrals E.Brownian functionals as stochastic integrals 3.5.The Girsanov Theorem A.The basic result B.Proof and ramifications C.Brownian motion with drift D.Thc Novikov condition 3.6.Loeal Time and a Generalized Ito Rule for Brownian Motion A.Definition of local time and the Tanaka formula B.The Trotter existence theorem C.Reflected Brownian motion and the Skorohod equation D.A generalized Ito rule for convex functions E.The Engelbert-Schmidt zero.one law 3.7.Local Time for Continuous Semimartingales 3.8.Solutions to Selected Problems 3.9.Notes CHAPTER 4 Brownian Motion and Partial Differential Equations 4.1.IntroductiOn 4.2.Harmonic Functions and the Dirichlet Problem A.The mean-value propcrty B.The Dirichlet problem C.Conditions for regularity D.Integral formulas of Poisson E.Supplementary exercises 4.3.The One.Dimensional Heat Equation A.The Tychonoff uniqueness theorem B.Nonnegative solutions of the heat equation
C.Boundary crossing probabilities for Brownian motion D.Mixed initial／boundary value problems 4.4.The Formulas of Fcynman and Kac A.The multidimensional formula B.The one.dimensional formula 4.5.Solutions to selected problems 4.6.Notes CHAPTER 5 Stochastic Differential Equations 5.1.IntroductiOn 5.2.Strong Solutions A.Definitions B.The It6 theory C.Comparison results and other refinements D.Approximations of stochastic differential equations E.Supplementary exercises 5.3.Weak Solutions A.Two notions of uniqueness B.Weak solutions by means of the Girsanov theorem C.A digression on regular conditional probabilities D.Results of Yamada and Watanabe on weak and strong solutions 5.4.The Martingale Problem of Stroock and Varadhan A.Some fundamental martingales B.Weak solutions and martingale problems C.Well-posedness and the strong Markov property D.Ouestions of existence E.Questions of uniqueness F.Supplementary exercises 5.5.A Study of the One-Dimensional Case A.The method of time change B.The method of removal of drift C.Feller』S test for explosions D.Supplementary exercises 5.6.Linear Equations A.GaUSS-Markov processes B.Brownian bridge C.The general，one-dimensional，linear equation D.SupDlementary exerciseS 5.7.Connections with PartiaI Difrerential Equations A.The Dirichlet problem B.The Cauchy problem and a Feynman—Kac representation C.Supplementary exercises 5.8.Applications to Economics A.Portfolio and consumption processes B.Option pricing C.Optimal consumption and investment(general theory) D.0ptimal consumption and investment(constant COcllidents) 5.9.Solutions to Selected Problems 5.10.Notes CHAPTER 6
P.Levy's Theory of Brownian Local Time 6.1.Introduction 6.2.AIternate Representations of Brownian Local Time A.The process of passage times B.Poisson random measures C.Subordinators D.The process of passage times revisited E.The excursion and downcrossing representations of local time 6.3.Two Independent Reflected Brownian Motions A.The positive and negative Darts of a Brownian motion B.The first formula of D.Williams C.The joint density of(W(t)，L(f)，Γ+(t)) 6.4.Elastic Brownian Motion A.The Feynman-Kac formulas for elastic Brownian motion B.The Ray-Knight description of local time C.The second formula of D.Williams 6.5.An Application：Transition Probabilities of Brownian Motion with Two-Valued Drift 6.6.Solutions to Selected Problems 6.7.NOtes Bibliography Index
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