An Advanced Course in Probability and Stochastic Processes provides a modern and rigorous treatment of probability theory and stochastic processes at an upper undergraduate and graduate level. Starting with the foundations of measure theory, this book introduces the key concepts of probability theory in an accessible way, providing full proofs and extensive examples and illustrations. Fundamental stochastic processes such as Gaussian processes, Poisson random measures, Lévy processes, Markov processes, and Itō processes are presented and explored in considerable depth, showcasing their many interconnections. Special attention is paid to martingales and the Wiener process and their central role in the treatment of stochastic integrals and stochastic calculus. This book includes many exercises, designed to test and challenge the reader and expand their skillset. An Advanced Course in Probability and Stochastic Processes is meant for students and researchers who have a solid mathematical background and who have had prior exposure to elementary probability and stochastic processes.
Key Features:
Focus on mathematical understanding Rigorous and self-contained Accessible and comprehensive High-quality illustrations Includes essential simulation algorithms Extensive list of exercises and worked-out examples Elegant and consistent notation
1. Measure Theory
2. Probability
3. Convergence
4. Conditioning
5.
Martingales
6. Wiener and Brownian Motion Processes
7. Itō Calculus Appendix
A. Selected Solutions Appendix B. Function Spaces Appendix C. Existence of
the Lebesgue Measure Index
Dirk P. Kroese, PhD, is a Professor of Mathematics and Statistics at The University of Queensland. He has published over 140 articles and seven books in a wide range of areas in applied probability, mathematical statistics, data science, machine learning, and Monte Carlo methods. He is a pioneer of the well-known Cross-Entropy methodan adaptive Monte Carlo technique, which is being used around the world to help solve difficult estimation and optimization problems in science, engineering, and finance.
Zdravko Botev, PhD, teaches Computational Statistics and Applied Probability at the University of New South Wales in Sydney, Australia. He is the recipient of the 2018 Christopher Heyde Medal of the Australian Academy of Science for distinguished research in the Mathematical Sciences and the 2019 Gavin Brown prize for his work on kernel density estimation, for which he is the author of one of the most widely used Matlab software scripts.
