布朗运动和随机计算（第2版）（英文影印版）

　　本书是Springer《数学研究生丛书》之113卷，是国内外公认的金融数学经典教材，各章有习题详解。本书初版于1988年，1991年出第2版，之后Springer已重印8次,本书是2005年的第8次重印版。...
　　

preface
suggestions for the reader
interdependence of the chapters
frequently used notation
chapter 1
martingales, stopping times, and filtrations
1.1. stochastic processes and (y-fields
1.2. stopping times
1.3. continuous-time martingales
1.4. the doob-meyer decomposition
1.5. 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
2.3. second construction of brownian motion
2.4. the space c[0, ∞), weak convergence, and wiener measure
2.5. the markov property

　　Two of the most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property.* This book is written for readers who are acquainted with both of these ideas in the discrete-time setting, and who now wish to explore stochastic processes in their continuoustime context. It has been our goal to write a systematic and thorough exposition of this subject, leading in many instances to the frontiers of knowledge.At the same time, we have endeavored to keep the mathematical prerequisites as low as possible, namely, knowledge of measure-theoretic probability and some familiarity with discrete-time processes. The vehicle we have chosen for this task is Brownian motion, which we present as the canonical example of both a Markov process and a martingale. We support this point of view by showing how, by means of stochastic integration and random time change,all continuous-path martingales and a multitude of continuous-path Markov processes can be represented in terms of Brownian motion. This approach forces us to leave aside those processes which do not have continuous paths.Thus, the Poisson process is not a primary object of study, although it is developed in Chapter 1 to be used as a tool when we later study passage times and local time of Brownian motion. .
　　The text is organized as follows: Chapter 1 presents the basic properties of marting ales, as they are used throughout the book. In particular, we generalizefrom the discrete to the continuous-time context the martingale convergence theorem, the optional sampling theorem, and the Doob-Meyer decomposition. The latter gives conditions under which a submartingale can be written
　　* According to M. Lo,ye, "martingales, Markov dependence and stationarity are the only three dependence concepts so far isolated which are sufficiently general and sufficiently amenable to investigation, yet with a great number of deep properties" (Ann. Probab. 1 (1973), p. 6).
　　as the sum of a martingale and an increasing process, and associates to every martingale with continuous paths a "quadratic variation process." This process is instrumental in the construction of stochastic integrals with respect to continuous martingales.
　　Chapter 2 contains three different constructions of Brownian motion, as well as discussions of the Markov and strong Markov properties for continuous-time processes. These properties are motivated by d-dimensional Brownian motion, but are developed in complete generality. This chapter also contains a careful discussion of the various filtrations commonly associated with Brownian motion. In Section 2.8 the strong Markov property is applied to a study of one-dimensional Brownian motion on a half-line, and on a bounded interval with absorption and reflection at the endpoints. Many densities involving first passage times, last exit times, absorbed Brownian motion, and reflected Brownian motion are explicitly computed. Section 2.9 is devoted to a study of sample path properties of Brownian motion. Results found in most texts on this subject are included, and in addition to these, a complete proof of the Levy modulus of continuity is provided.
　　The theory of stochastic integration with respect to continuous martingales is developed in Chapter 3. We follow a middle path between the original constructions of stochastic integrals with respect to Brownian motion and the more recent theory of stochastic integration with respect to right-continuous martingales. By avoiding discontinuous martingales, we obviate the need to introduce the concept of predictability and the associated, highly technical,measure-theoretic machinery. On the other hand, it requires little extra effort to consider integrals with respect to continuous martingales rather than merely Brownian motion. The remainder of Chapter 3 is a testimony to the power of this more general approach; in particular, it leads to strong theorems concerning representations of continuous martingales in terms of Brownian motion (Section 3.4). In Section 3.3 we develop the chain rule for stochastic calculus, commonly known as It6's formula. The Girsanov Theorem of Section 3.5 provides a method of changing probability measures so as to alter the drift of a stochastic process. It has become an indispensable method for constructing solutions of stochastic differential equations (Section 5.3) and is also very important in stochastic control (e.g., Section 5.8) and filtering. Local time is introduced in Sections 3.6 and 3.7, and it is shown how this concept leads to a generalization of the It6 formula to convex but not necessarily differentiable functions. ..
　　Chapter 4 is a digression on the connections between Brownian motion,Laplace's equation, and the heat equation. Sharp existence and uniqueness theorems for both these equations are provided by probabilistic methods;applications to the computation of boundary crossing probabilities are discussed, and the formulas of Feynman and Kac are established.
　　Chapter 5 returns to our main theme of stochastic integration and differential equations. In this chapter, stochastic differential equations are driven by Brownian motion and the notions of strong and weak solutions are presented. The basic It6 theory for strong solutions and some of its ramifications, including comparison and approximation results, are offered in Section 5.2, whereas Section 5.3 studies weak solutions in the spirit of Yamada & Watanabe. Essentially equivalent to the search for a weak solution is the search for a solution to the "Martingale Problem" of Stroock & Varadhan.In the context of this martingale problem, a full discussion of existence, uniqueness, and the strong Markov property for solutions of stochastic differ ential equations is given in Section 5.4. For one-dimensional equations it is possible to provide a complete characterization of solutions which exist only up to an "explosion time," and this is set forth in Section 5.5. This section also presents the recent and quite striking results of Engelbert & Schmidt concerning existence and uniqueness of solutions to one-dimensional equations.This theory makes substantial use of the local time materiat of Sections 3.6,3.7 and the martingale representation results of Subsections 3.4.A,B. By analogy with Chapter 4, we discuss in Section 5.7 the connections between solutions to stochastic differential equations and elliptic and para~olic partial differential equations. Applications of many of the ideas in Chapters 3 and 5 are contained in Section 5.8, where we discuss question~ of option pricing and optimal portfolio/consumption management. In particular, the Girsanov theorem is used to remove the difference between average rates of return of different stocks, a martingale representation result provides the optimal portfolio process, and stochastic representations of solutions to partial differential equations allow us to recast the optimal portfolio and consumption management problem in terms of two linear parabolic partial differential equations, for which explicit solutions are provided.
　　Chapter 6 is for the most part derived from Paul Levy's profound study of Brownian excursions. Levy's intuitive work has now been formalized by such notions as filtrations, stopping times, and Poisson random measures, but the remarkable fact remains that he was able, 40 years ago and working without these tools, to penetrate into the fine structure of the Brownian path and to inspire all the subsequent research on these matters until today. In the spirit of Levy's work, we show in Section 6.2 that when one travels along the Brownian path with a clock run by the local time, the number of excursions away from the origin that one encounters, whose duration exceeds a specified number, has a Poisson distribution. L6vy's heuristic construction of Brownian motion from its excursions has been made rigorous by other authors. We do not attempt such a construction here, nor do we give a complete specification of the distribution of Brownian excursions; in the interest of intelligibility, we content ourselves with the specification of the distribution for the durations of the excursions. Sections 6.3 and 6.4 derive distributions for functionals of Brownian motion involving its local time; we present, in particular, Feynman-Kac result for the so-called "elastic" Brownian motion, the for mulas of D. Williams and H. Taylor, and the Ray-Knight description of Brownian focal time. An application of this theory is given in Section 6.5,where a one-dimensional stochastic control problem of the "bang-bang" type is solved.
　　The writing of this book has become for us a monumental undertaking involving several-people, whose assistance we grafefully acknowledge..Foremost among these are the member of our families, Eleni, Dot, Andrea,and Matthew, whose support, encouragement, and patience made the whole endeavor possible. Parts of the book grew out of notes on lectures given at Columbia University over several years, and we owe much to the audiences in those courses. The inclusion of several exercises, the approaches taken to a number of theorems, and several citations of relevant literature resulted from discussions and correspondence with F. Baldursson, A. Dvoretzky,W. Fleming, O. Kallenberg, T. Kurtz, S. Lalley, J. Lehoczky, D. Stroock, and M. Yor. We have also taken exercises from Mahdi, Lanska & Vrkotc (1978),and Ethier & Kurtz (1986). As the project proceeded, G.-L. Xu, Z.-L. Ying,and Th. Zariphopoulou read large portions of the manuscript and suggested numerous corrections and improvements. Careful reading by Daniel Ocone and Manfred Schiil revealed minor errors in the first printing, and these have been corrected. Others, including F. Akesson, S. Dayanik, B. Doytchinov,H.J. Engelbert, R. H6hnle, C. Hou, A. Karolik, W. Nichols, L. Nielsen, D.Ocone, N. Vaillant and H. Wang found errors and/or contributed ideas,which have resulted in improvements in subsequent printings. However, our greatest single debt of gratitude goes to Marc Yor, who read much of the near-final draft and offered substantial mathematical and editorial comments on it. The typing was done tirelessly, cheerfully, and efficiently by Stella DeVito and Doodmatie Kalicharan; they have our most sincere appreciation.
　　We are grateful to Sanjoy Mitter and Dimitri Bcrtsekas for extending to us the invitation to spend the critical initial year of this project at the Massachusetts Institute of Technology. During that time the first four chapters were essentially completed, and we were partially supported by the Army Research Office under grant DAAG-299-84-K-0005. Additional financial support was provided by the National Science Foundation under grants DMS-84-16736 and DMS-84-03166 and by the Air Force Office of Scientific Research under grants AFOSR 82-0259, AFOSR 85-0360, and AFOSR 86-0203. ...
　　Ioannis Karatzas
　　Steven E. Shreve