Ntroduction to Stochastic Processes Erhan Cinlar

Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an introduction to the theory of martingales and semimartingales, random measures stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. The second edition contains some additions to the text and references. Some parts are completely rewritten.

This book focuses on the modeling and mathematical analysis of stochastic dynamical systems along with their simulations. The collected chapters will review fundamental and current topics and approaches to dynamical systems in cellular biology.

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.