Stochastic numerical methods

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This book presents advanced numerical methods utilizing probability and stochastic concepts to analyze random processes. Targeted at students in natural sciences—such as Physics, Chemistry, Mathematics, and Biology—as well as Engineering and social sciences like Economics and Sociology, it highlights techniques recently applied to agent-based models. The authors provide detailed examples from the field of phase transitions, illustrating the entire process from numerical simulation and algorithm design to data analysis, including critical exponent extraction and finite-size effects. Central to the text are Monte Carlo methods, with applications in statistical physics and phase transitions, alongside numerical methods for both ordinary and partial stochastic differential equations, featuring advanced pseudo-spectral methods. Gillespie's method is also discussed for simulating systems governed by master equations, with applications in Biology, such as protein expression and transcription. To elucidate modern hybrid algorithms that integrate Monte Carlo and stochastic differential equations, the authors introduce the fundamentals of molecular dynamics. The book includes appendices with supplementary material on advanced topics, practical exercises at the end of each chapter, and useful codes for the core methods.