deterministic motion. Further advances in this line were made by Leonhard Euler and William Rowan Hamilton, and, in 1778, Joseph-Louis Lagrange proposed a new formulation of classical mechanics. This new formulation is based on the optimization of energy functionals, and it allows us to solve more sophisticated motion problems in a more systematic manner. This new approach to mechanics is the basis for dealing with modern quantum mechanics and the physics of high-energy particles. Albert Einstein, in 1905, through his special theory of relativity, introduced fine corrections for high velocities, close to the maximum speed of light. All of these fundamental mathematical models deal with many sophisticated motions such as that of astronomical objects, satellites (natural and artificial), particles in intense electromagnetic fields, particles under gravitational forces, a better understanding of light, and so on. However, even though these might be very complicated problems, we should mention that all of them have deterministic paths of motion.
In 1827, the Scottish botanist Robert Brown described a special kind of random motion produced by the interaction of many particles, while looking at pollen of the plant Clarkia pulchella immersed in water, using a microscope, and it was recognized that this type of motion could not be fully explained by modeling the motion of each particle or molecule. Albert Einstein, 78 years later, in 1905, published a seminal paper where he modeled the motion of the pollen as being moved by individual water molecules. In this work, the diffusion equation was introduced as a convenient mathematical model for this random phenomenon. A related model in this direction was presented in 1906 by Marian Smoluchowski, and the experimental verification was done by Jean Baptiste Perrin in 1908.
A similar model for Brownian motion was proposed in 1900 by Louis Bachelier in his PhD thesis entitled The Theory of Speculation, where he presented a stochastic analysis for valuing stock options in financial markets. This novel application of a stochastic model faced criticism at the beginning, but Bachelier’s instructor Henri Poincaré was in full support of this visionary idea. This fact shows a close relationship between models to explain random phenomena in physics (statistical mechanics) and in financial analysis, and also in many other areas.
A notable contribution of the American mathematician Norbert Wiener was to establish the mathematical foundations for Brownian motion, and for that reason it is also known as the Wiener process. Great mathematicians such as Paul Lévy, Andrey Kolmogorov and Kiyosi Itô, among many other brilliant experts in the new field of probability and stochastic processes, set the basis of these stochastic processes. For example, the famous Black–Scholes formula in financial markets is based on both diffusion processes and Itô’s ideas.
In spite of its success in modeling many types of random motion and other random quantities, the Wiener process has some drawbacks when capturing the physics of many applications. For instance, the modulus of velocity is almost always infinite at any instant in time, it has a free path length of zero, the path function of a particle is almost surely non-differentiable at any given point and its Hausdorff dimension is equal to 1.5, i.e. the path function is fractal. However, the actual movement of a physical particle and the actual evolution of share prices are barely justified as fractal quantities. Taking into account these considerations, in this book we propose and develop other stochastic processes that are close to the actual physical behavior of random motion in many other situations. Instead of the diffusion process (Brownian motion), we consider telegraph processes, Poisson and Markov processes and renewal and semi-Markov processes.
Markov (and semi-Markov) processes are named after the Russian mathematician Andrey Markov, who introduced them in around 1906. These processes have the important property of changing states under certain rules, i.e. they allow for abrupt changes (or switching) in the random phenomenon. As a result, these models are more appropriate for capturing random jumps, alternate velocities after traveling a certain random distance, random environments through the formulation of random evolutions, random motion with random changes of direction, interaction of particles with non-zero free paths, reliability of storage systems, and so on.
In addition, we will model financial markets with Markov and semi-Markov volatilities as well as price covariance and correlation swaps. Numerical evaluations of variance, volatility, covariance and correlations swaps with semi-Markov volatility are also presented. The novelty of these results lies in the pricing of volatility swaps in the closed form, and the pricing of covariance and correlation swaps in a market with two risky assets.
Anatoliy POGORUI
Zhytomyr State University, Ukraine
Anatoliy SWISHCHUK
University of Calgary, Canada
Ramón M. RODRÍGUEZ-DAGNINO
Tecnologico de Monterrey, Mexico
October 2020
Acknowledgments
Anatoliy Pogorui was partially supported by the State Fund for Basic Research (Ministry of Education and Science of Ukraine 20.02.2017 letter no. 12).
Anatoliy Pogorui
I would like to thank NSERC for its continuing support, my research collaborators, and my current and former graduate students. I also give thanks to my family for their inspiration and unconditional support.
Anatoliy Swishchuk
I would like to thank Tecnologico de Monterrey for providing me the time and support for these research activities. I also appreciate the support given by Conacyt through the project no. SEP-CB-2015-01-256237. The time and lovely support given to me by my wife Saida, my three daughters Dunia, Melissa and R. Melina, as well as my son Ramón Martín, are invaluable.
Ramón M. Rodríguez-Dagnino
Introduction
I.1. Overview
The theory of dynamical systems is one of the fields of modern mathematics that is under intensive study. In the theory of stochastic processes, researchers are actively studying dynamical systems, operating under the influence of random factors. A good representative of such systems is the theory of random evolutions. The first results within this field were obtained by Goldstein (1951) and Kac (1974), who studied the movement of a particle on a line with a speed that changes its sign under the Poisson process. Subsequently, this process was called the telegraph process or the Goldstein–Kac process. Further developments of this theory have been presented in the works of Griego and Hersh (1969, 1971), Hersh and Pinsky (1972), and Hersh (1974, 2003), which gave a definition of stochastic evolutions in a general setting.
Important advances in the theory of stochastic evolutions have been made in the formulation of limit theorems and their refinement; these consist of obtaining asymptotic expansions. These problems are studied in the works of Korolyuk and Turbin (1993), Skorokhod (1989), Turbin (1972, 1981), Korolyuk and Swishchuk (1986), Korolyuk and Limnios (2009, 2005), Shurenkov (1989, 1986), Dorogovtsev (2007b), Anisimov (1977), Girko (1982), Hersh and Pinsky (1972), Papanicolaou (1971a,b), Kertz (1978), Watkins (1984, 1985), Balakrishnan et al. (1988), Yeleyko and Zhernovyi (2002), Pogorui (1989, 1994, 2009a) and Pogorui and Rodríguez-Dagnino (2006, 2010a).
Among the many methods in finding limiting theorems, we should mention a class that could be called an asymptotic average scheme. By using these methods, a semi-Markov evolution can be reduced to a random evolution in a lumping state space with Markov switching, and is thus studied as a Markov scheme. Most of
