An Introduction to Continuous-Time Stochastic Processes

This book is a systematic, rigorous, and self-consistent introduction to the theory of continuous-time stochastic processes. But it is neither a tract nor a recipe book as such; rather, it is an account of fundamental concepts as they appear in relevant modern applications and literature. We make no pretense of it being complete. Indeed, we have omitted many results, which we feel are notdirectly relatedtothemain themeorthatare availablein easilyaccessible sources. (Thosereaderswhoareinterestedinthehistoricaldevelopmentofthe subject cannot ignore the volume edited by Wax (1954). ) Proofs are often omitted as technicalities might distract the reader from a conceptual approach. They are produced whenever they may serve as a guide to the introduction of new concepts and methods towards the app- cations; otherwise, explicit references to standard literature are provided. A mathematically oriented student may ?nd it interesting to consider proofs as exercises. The scope of the book is profoundly educational, related to modeling re- world problems with stochastic methods. The reader becomes critically aware oftheconceptsinvolvedincurrentappliedliterature,andismoreoverprovided with a ?rm foundation of the mathematical techniques. Intuition is always supported by mathematical rigor. Our book addresses three main groups: ?rst, mathematicians working in a di?erent ?eld; second, other scientists and professionals from a business or academic background; third, graduate or advanced undergraduate students of a quantitative subject related to stochastic theory and/or applications.

"This is an introductory text on continuous time stochastic processes and their applications to finance and biology. The first part of the book reviews basic probability and then covers the basic continuous time processes such as Brownian motion, point processes, etc.... The book will be useful for applied mathematicians who are not probabilists to get a quick flavour of the techniques of stochastic calculus, and for professional probabilists to get a quick flavour of the applications."   —Mathematical Reviews

"The book is a systematic, rigorous, and self-contained introduction to the theory of continuous-time stochastic processes. It is an account of fundamental concepts as they appear in relevant modern applications and literature.... The book addresses three main groups: first, mathematicians working in a different field; second, other scientists and professionals from a business or academic background; third, graduate or advanced undergraduate students of a quantitative subject related to stochastic theory and/or applications."   —Zentralblatt MATH

"This book provides a mathematical overview of the theory of continuous-time stochastic processes, with emphasis on stochastic differential equations (SDEs). Applications in finance and population modelling are also briefly reviewed.... The primary audience for this book will be mathematicians (both pure and applied) active in other areas who require an introduction to stochastic theory. Scientists already working in the applications of SDEs will also benefit from this mathematically rigorous reference text. The core of the text on Ito calculus...would be suitable supplementary reading for graduate or advanced undergraduate students of stochastic theory.... The style of the text...is concise and rigorous.... Each chapter concludes with a set of exercises inviting readers to prove supplementary results and review particular aspects of thetheory.... In summary, I have found this to be a useful reference text, and would recommend it to those wishing to delve into the mathematical theory of stochastic processes."   —UK Nonlinear News

"This book covers an extensive part of probability theory, the theory for time-continuous stochastic processes, and also gives a lot of examples from finance, biology, and medicine.... Most of the contents of the book have been used as lecture material for several years, which is evident by the fact that there are very few misprints, and the text is well-structured also having a good labelling system.... [T]he book is quite compact covering the fundamentals in probability theory and stochastic processes in 200 pages.... The chapter on applications towards finance covers many classical models such as arbitrage-free markets, [the] Black–Scholes model, and models for interest and ruin probabilities. This chapter is probably the best in the book.... The chapter on applications towards biology and medicine…contains models for epidemics, individual-based models, and a model for neural activity.... After each chapter there are several exercises illustrating and extending the theory presented in the text. Many of these exercises are interesting and rewarding to solve."   —Mathematical Biosciences

"The book is written in a systematic and self-contained way where omitted details are compensated by references to a commonly accessible literature. The book is [geared] to students or professionals who want to get acquainted with the role of stochastic processes in modeling random phenomena in economics, biology, or medicine." —Applications of Mathematics