![]() Starting from this result one can expand the set of equations that can be solved by using Girsanov’s formula to change the drift \(b\) and by considering weak solutions. This approach dates back to Itô in the 1940s. When the coefficients \(b\) and \(\sigma\) are Lipschitz then the SDE at the beginning of this review can be solved by Picard iteration. This listing may not mean much to people who don’t know the subject but at least you can see that its development required the efforts of a number of famous probabilists.Ĭhapters 15–18 develop the theory of stochastic differential equations. ![]() The Doob-Meyer decomposition, Kunita-Watanable inequality, Burkholder-David-Gundy inequality and Itô’s formula are proved. The next four chapters give the heart of the theory: semimartingales and the quadratic variation are defined. Since ordinary integration is a special case it begins with a discussion of integration with respect to processes of finite variation, i.e., signed measures. This topic is somewhat delicate because there are uncountably many times, so if there is a null set for each one the union can have positive probability.Ĭhapters 8–14 develop stochastic integration. The various versions of the notion that \(H(s)\) can be determined from what is known at time \(s\) (optional and predictable process) are the subject of chapter 7. Of course, for mathematical and legal reasons, \(H(s)\) cannot be allowed to look at the future beyond time \(s\). Then the stochastic integral \ gives our profit from time \(0\) to time \(t\). To explain the issues at hand, suppose we are given a martingale \(M(s)\), which we are thinking of as a stock price, and a process \(H(s)\) corresponding to the number of shares we hold at time \(s\). A review of the first edition quoted on the book’s cover says that the book can be recommended for graduate students, but I think many of them will have difficulties with chapters 3, 6, and 7, which discuss the measure theoretic difficulties that arise in properly formulating the stochastic integral. Chapters 4 and 5 discuss martingales in discrete and continuous time, essential prerequisites for building the theory. The journey begins with a two chapter introduction to measure theoretic probability. To quote the authors “the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integration.” Much has happened in the last 30+ years, so the book has grown from 300 to 625 pages. The beginnings of that subject can be traced to Paul André Meyer’s 1976 article in Séminare des Probabilités de Strasbourg, an annual series of books published in Springer’s Lecture Notes in Mathematics series in the 1960s and 1970s. ![]() This is the second edition of a book first written in 1982 when the general theory of stochastic integration was in its infancy. Here the authors concentrate on applications to optimal control and filtering, but there are 19 chapters to be covered before one reaches the applications. These processes arise as large population limits in population genetics and epidemiology, but their most famous use is in mathematical finance to model stock prices. We think of this as the differential equation \(dX_t/dt =b(X_t)\) perturbed by noise (which may or may not be small). One of the basic objects of study is the stochastic differential equation (SDE), which in its simplest form can be written as \ where \(B_t\) is a one-dimensional Brownian motion. Stochastic calculus is such a broad subject that it is hard to describe.
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