This book accompanies the reader from the simple, standard Poisson process to its more complex transformations, using a unified framework, showing proofs of basic theorems and references to more difficult results.

The reader will also benefit from the empirical applications of the various models to real data in finance and insurance (e.g., stock market crashes and fire accidents), physics and geology (e.g., earthquakes), biology and demography (e.g., births, deaths, and pandemics), and more. Many of the applications relate to financial mathematics, and a professional can glean a lot from an understanding of how to apply this important mathematical tool.

Some examples are worked out in open-source R showing the reader how to implement the models. Strengths of Poisson Process and its Fractional Extensions with Applications are:

• A focus on a very fundamental class of stochastic processes
• An introductory approach assuming no previous experience with the topics
• A wide range of generalizations considered, including recent advances to which one of the authors has made seminal contributions
• Attention not only to theory, but also to practical applications with real-data examples from a wide range of fields

The authors pay special attention not only to the theoretical foundations of the Poisson process and the various generalizations that have been proposed in the literature in the last decades, but also to the practical applications in many different fields of the Poisson models for a better understanding of the stochastic structure of the real world.

Note that for the sake of the reader, the probability generating functions are summarized in Appendix A; subordinators in Appendix B; fractional derivatives in Appendix C; gamma and Mittag-Leffler functions in Appendix D; and tools for data analysis in Appendix E. Appendix F contains the R code.