Telegraph Processes and Option Pricing

This book provides an extensive, systematic overview of the modern theory of telegraph processes and their multidimensional counterparts, together with numerous fruitful applications in financial modelling. Focusing on stochastic processes of bounded variation instead of classical diffusion, or more generally, Lévy processes, has two obvious benefits. First, the mathematical technique is much simpler, which helps to concentrate on the key problems of stochastic analysis and applications, including financial market modelling. Second, this approach overcomes some shortcomings of the (parabolic) nature of classical diffusions that contradict physical intuition, such as infinite propagation velocity and infinite total variation of paths. In this second edition, some sections of the previous text are included without any changes, while most others have been expanded and significantly revised. These are supplemented by predominantly new results concerning piecewise linear processes with arbitrary sequences of velocities, jump amplitudes, and switching intensities. The chapter on functionals of the telegraph process has been significantly expanded by adding sections on exponential functionals, telegraph meanders and running extrema, the times of the first passages of telegraph processes with alternating random jumps, and distribution of the Euclidean distance between two independent telegraph processes. A new chapter on the multidimensional counterparts of the telegraph processes is also included.

The book is intended for graduate students in mathematics, probability, statistics and quantitative finance, and for researchers working at academic institutions, in industry and engineering. It can also be used by university lecturers and professionals in various applied areas.

Alexander Kolesnik is a Professor, former Head of Laboratory (2015-2019) and Principal Researcher (since 2020) at the Institute of Mathematics and Computer Science ＂Vladimir Andrunachievici,＂ Kishinev (Chișinău), Moldova. He graduated from Moldova State University (1975-1980) and pursued his postgraduate studies (1987-1991) at the Institute of Mathematics of the National Academy of Sciences of Ukraine. He earned his PhD (1991) and PhD Habilitation (2010) degrees in mathematics and physics with a specialization in stochastic processes, probability and statistics, conferred by the Institute of Mathematics of the National Academy of Sciences of Ukraine. His research interests include: probability and statistics, stochastic processes, random evolutions, stochastic dynamic systems, random flights, diffusion processes, transport processes, random walks, stochastic processes in random environments, partial differential equations in stochastic models, statistical physics, and wave processes. Dr Kolesnik has authored more than 70 scientific publications, predominantly in leading international journals, and two monographs. He has also served as an external reviewer for many outstanding international journals in mathematics and physics, garnering him e.g. a ＂Certificate of Outstanding Contribution in Reviewing＂ from the journal ＂Stochastic Processes and their Applications.＂ He has been a visiting professor and scholarship holder at several universities in Italy and Germany and served on the Board of Global Advisors of the International Federation of Nonlinear Analysts (IFNA), USA.

Nikita Ratanov graduated from Moscow State University (Lomonosov) in 1976. He subsequently earned his PhD (1984) at the same institution, followed by a doctorate in PDE and Stochastics (1999). He has published over 100 papers, two textbooks (in Russian and Spanish) and a monograph in these areas. For nearly the past two decades, he has been a Professor at the Universidad del Rosario (Bogotá, Colombia). He is currently affiliated with Chelyabinsk State University (Russia) as a Leading Research Fellow. Dr Ratanov has a wide range of research interests, including stochastic analysis with applications in statistical physics, financial market modelling and some aspects of neural modelling.