Anomalous Stochastics: A Comprehensive Guide to Multifractals, Random Walks, and Real-World Applications

This textbook provides a comprehensive exploration of anomalous stochastic processes and extreme events, commonly referred to as "black swans," with a particular focus on (multi-)fractal approaches and continuous-time random walks. The authors present a systematic examination of the subject, tracing its inception and providing a multi-directional perspective. By drawing on real-world experiences in finance, physics, and technology, the book underscores the practical relevance of anomalous stochastic processes for practitioners dealing with real-world data from complex systems.

The content is based on a series of interdisciplinary physics lectures that have been delivered to undergraduate and graduate students at the University of Warsaw for nearly two decades. Updated to reflect recent developments, this book is a valuable resource for graduate students, ambitious undergraduate students, and researchers interested in random processes and the practical implications of anomalous processes. Familiarity with fundamental principles of probability theory, algebra, and basic concepts of differential and integral calculus is assumed, while a foundational understanding of mathematical statistics, stochastic processes, and statistical thermodynamics is recommended.

Additionally, each chapter includes practical exercises designed to help readers master the concepts, develop practical skills, and serve as teaching material.

Michal Chorowski is a researcher and quantitative analyst with a robust background in physics, psychology, and computational modeling. Currently, he serves as a Quantitative User Experience (UX) Researcher at Google, where he applies his expertise in data analysis and quantitative methods to measure and evaluate the user experience of digital products and services.

Simultaneously, Michal is pursuing a PhD in Physics at the Faculty of Physics University of Warsaw, under the supervision of Professor Ryszard Kutner. His academic research focuses on modeling critical phenomena in economic markets, time series forecasting, and multifractal analysis. Through his work, he aims to contribute to a deeper understanding of market behavior and forecasting methodologies.

Michal has gained valuable experience in the finance industry while working as a Projections Modeller in the Quantitative Strategies Team at Credit Suisse. In this capacity, he developed mathematical models for scenario-based stress testing and risk assessments, thereby enhancing risk management practices. He holds an MSc in Psychology from the University of Warsaw, where he explored the psychological factors influencing product trends and forecasting in his thesis. Additionally, he earned an MSc in Physics from the Faculty of Physics at the University of Warsaw, specializing in Econophysics, with a focus on modeling government intervention in technology development and innovation diffusion. His exceptional work was recognized when he received the first prize for the best Master's thesis from the Physics in Economics and Social Sciences (FENS) section of the Polish Physical Society in 2021.

Michal Chorowski's research findings have been published in international journals and presented at various academic conferences, contributing to the advancement of knowledge in his fields of expertise.

Tomasz Gubiec earned his MSc in Physics from the University of Warsaw in 2007, followed by a PhD in Physics from the same institution in 2011. He was awarded a Fulbright Scholarship, which enabled him to pursue his studies at Boston University. Additionally, he received the Foundation for Polish Science START Scholarship, further supporting his research endeavors. During his time at Boston University, Tomasz had the privilege of working under the guidance of Professor Eugene Stanley. His research focused on stochastic processes and econophysics, integrating principles from physics and economics to gain insights into complex financial systems.

Tomasz's expertise and interdisciplinary approach led him to assume a leadership role in an interdisciplinary research group, where he extensively analyzed the stability of the Polish interbank market. Currently, he holds the position of Assistant Professor at the Faculty of Physics University of Warsaw. In addition to his academic responsibilities, he also serves as a consultant for hedge funds, leveraging his in-depth understanding of financial markets and stochastic processes to provide valuable insights and guidance.

Tomasz's research findings have been widely recognized and disseminated through his publication of over 20 research articles in reputable scientific journals. His work has made significant contributions to the fields of stochastic processes, econophysics, and agent-based modeling in finance. Committed to advancing scientific knowledge, Tomasz is also passionate about popularizing science among the general public. His efforts in this regard were acknowledged when he received the first prize in the  popularization of science competition INTER, organized by the Foundation for Polish Science.

With 15 years of experience in teaching stochastic processes, Tomasz has played a crucial role in shaping the education of aspiring physicists and researchers in this specialized field. His expertise, combined with his passion for teaching, has helped cultivate a new generation of scholars equipped with a solid understanding of stochastic processes.

Ryszard Kutner earned his MSc in Physics from the Faculty of Physics at the University of Warsaw and completed his PhD at the same institution in 1977. From 1979 to 1981, he held a postdoctoral position in Professor Kurt Binder’s group at the Institut für Festkörperforschung at Forschungszentrum Jülich in Germany. Following his postdoctoral work, he returned to the Faculty of Physics at the University of Warsaw, where he served as an Assistant Professor and obtained his Habilitation in 1985. His passion for teaching and research led him to assume various academic roles, including a visiting professorship in the Department of Physics at Konstanz University in Germany from 1993 to 1995, and a scientific researcher position in the Department of Applied Physics at the Tokyo Institute of Technology in Japan in 1991.

Dr. Kutner's research primarily focuses on analyzing extreme („black swan”) and super-extreme (dragon king) events in stochastic dynamics, making significant contributions to the field. His exceptional work was recognized through the direction of a prestigious research grant (under The Economic Research Framework Programme of the National Bank of Poland for the years 2009-2012), which was distinguished by the President of the National Bank of Poland. As a highly accomplished author, Dr. Kutner has published numerous research articles, review articles, and topical issues covering a wide range of topics, including hierarchical spatiotemporal coupling, extreme events, and multifractality. His publications have solidified his position as a leading authority in the field of econophysics.

Additionally, he serves as the elected chairperson of the Physics in Economics and Social Sciences Section of the Polish Physical Society and is a member of the society's scientific awards committee. Since 2004, this section has been organizing twelve symposiums.