Uantum Stochastic Processes And Noncommutative Geometry

This textbook gives a comprehensive introduction to stochastic processes and calculus in the fields of finance and economics, more specifically mathematical finance and time series econometrics. Over the past decades stochastic calculus and processes have gained great importance, because they play a decisive role in the modeling of financial markets and as a basis for modern time series econometrics. Mathematical theory is applied to solve stochastic differential equations and to derive limiting results for statistical inference on nonstationary processes.

Mathematics is publishing a Special Issue to honor Prof. Sally McClean on the occasion of her semi-retirement and in recognition of her important research contributions. Sally Ida McClean was born in Belfast and received her first degree, an M.A. in Mathematics, from the University of Oxford in 1970. She earned an M.Sc. in Mathematical Statistics and Operations Research from Cardiff University in 1972, and completed her Ph.D. in 1976 at Ulster University at Coleraine. Her contribution to mathematical modeling in healthcare planning is enormous, and, in particular, her studies on improving the wellbeing of the elderly are greatly respected amongst her peers. She is currently a Professor of Mathematics at Ulster University. Her main research interests are in Stochastic Modeling and Optimization for Healthcare Planning and Computer Science. Stochastic processes are some of the most important tools in many areas of science, such as biology, operational research, the social sciences, stochastic finance, etc. Important characteristics in these areas evolve with time in a relatively random way, and since stochastic processes are mainly sequences or families of random variables, in which their index represents time, they are the natural tool to use. The theory and applications of stochastic processes emerged in the genesis of one of the richest ones, that is, Brownian motion. This was rather unexpected since Brownian motion is a beautiful object which is at the same time a martingale, a Gaussian process, a diffusion, a Levy process, a Markov process, etc.—concepts that were discovered quite latter in the evolution of time.

This text on stochastic processes and their applications is based on a set of lectures given during the past several years at the University of California, Santa Barbara (UCSB). It is an introductory graduate course designed for classroom purposes. Its objective is to provide graduate students of statistics with an overview of some basic methods and techniques in the theory of stochastic processes. The only prerequisites are some rudiments of measure and integration theory and an intermediate course in probability theory. There are more than 50 examples and applications and 243 problems and complements which appear at the end of each chapter. The book consists of 10 chapters. Basic concepts and definitions are pro­ vided in Chapter 1. This chapter also contains a number of motivating ex­ amples and applications illustrating the practical use of the concepts. The last five sections are devoted to topics such as separability, continuity, and measurability of random processes, which are discussed in some detail. The concept of a simple point process on R+ is introduced in Chapter 2. Using the coupling inequality and Le Cam's lemma, it is shown that if its counting function is stochastically continuous and has independent increments, the point process is Poisson. When the counting function is Markovian, the sequence of arrival times is also a Markov process. Some related topics such as independent thinning and marked point processes are also discussed. In the final section, an application of these results to flood modeling is presented.