An Introduction to Continuous-Time Stochastic Processes

This textbook, now in its fourth edition, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, it features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required. Unlike other books on stochastic methods that specialize in a specific field of applications, this volume examines the ways in which similar stochastic methods can be applied across dierent elds.
Beginning with the fundamentals of probability, the authors go on to introduce the theory of stochastic processes, the Itô Integral, and stochastic differential equations. The following chapters then explore stability, stationarity, and ergodicity. The second half of the book is dedicated to applications to a variety of fields, including finance, biology, and medicine. Some highlights of this fourth edition include a more rigorous introduction to Gaussian white noise, additional material on the stability of stochastic semigroups used in models of population dynamics and epidemic systems, and the expansion of methods of analysis of one-dimensional stochastic dierential equations.
An Introduction to Continuous-Time Stochastic Processes, Fourth Edition is intended for graduate students taking an introductory course on stochastic processes, applied probability, stochastic calculus, mathematical finance, or mathematical biology. Prerequisites include knowledge of calculus and some analysis; exposure to probability would be helpful but not required since the necessary fundamentals of measure and integration are provided. Researchers and practitioners in mathematical finance, biomathematics, biotechnology, and engineering will also find this volume to be of interest, particularlythe applications explored in the second half of the book.

Introduces readers to the theory of continuous-time stochastic processes using real-life examples in medicine, finance, and biology Includes updated exercises, examples, and material based on advances in recent literature Illustrates the ways that similar stochastic methods can be applied broadly across different fields

Auteur

Vincenzo Capasso is a Professor of Probability and Mathematical Statistics at the University of Milan, an elected member of the International Statistics Institute, a Fellow of The Institute of Mathematics and its Applications - UK, Past President of ECMI (the European Consortium for Mathematics in Industry), and Past President of ESMTB (European Society for Mathematical and Theoretical Biology). David Bakstein has been working in the financial industry for close to 25 years, many of those dedicated to applied mathematical models. He originally studied and taught at both the LSE and University of Oxford (OCIAM & Lady Margaret Hall).

Résumé
"Exercises are provided at the end of each chapter; the difficulty ranges from basic applications to more advanced ideas ... . Overall this book is a nice way to get into the basics of stochastic processes for someone working in a different field. It is quite reasonable that this could serve as either a main textbook or secondary reference for a graduate course. Sufficient details on each topic are provided by the authors, which makes this possible." (Eric Stachura, MAA Reviews, January 30, 2022)

Contenu
Foreword.- Preface to the Fourth Edition.- Preface to the Third Edition.- Preface to the Second Edition.- Preface.- Part I: Theory of Stochastic Processes.- Fundamentals of Probability.- Stochastic Processes.- The Itô Integral.- Stochastic Differential Equations.- Stability, Stationary, Ergodicity.- Part II: Applications of Stochastic Processes.- Applications to Finance and Insurance.- Applications to Biology and Medicine.- Measure and Integration.- Convergence of Probability Measures on Metric Spaces.- Diffusion Approximation of a Langevin System.- Elliptic and Parabolic Equations.- Semigroups of Linear Operators.- Stability of Ordinary Differential Equations.- References.- Nomenclature.- Index.