Ambit Stochastics

Drawing on advanced probability theory, Ambit Stochastics is used to model stochastic processes which depend on both time and space. This monograph, the first on the subject, provides a reference for this burgeoning field, complete with the applications that have driven its development.

Unique to Ambit Stochastics are ambit sets, which allow the delimitation of space-time to a zone of interest, and ambit fields, which are particularly well-adapted to modelling stochastic volatility or intermittency. These attributes lend themselves notably to applications in the statistical theory of turbulence and financial econometrics. In addition to the theory and applications of Ambit Stochastics, the book also contains new theory on the simulation of ambit fields and a comprehensive stochastic integration theory for Volterra processes in a non-semimartingale context.

Written by pioneers in the subject, this book will appeal to researchers and graduate students interested in empirical stochastic modelling.

• 60G60, 60F05, 60H05, 60H07, 60H15, 60H20, 60J75, 62F12
• 62H11, 62M10, 62M30, 62P20, 62P35, 65C30, 76F55, 76M35
• 91B25, 91B70
• Volterra processes
• Lévy processes
• Ambit fields
• volatility/intermittency
• statistical turbulence
• power variation
• stochastic partial differential equations
• stochastic PDEs
• random fields
• Lévy basis
• energy markets
• stochastic integration
• non-semimartingales
• Trawl processes
• quantitative finance

“The author pays a great attention to diverse methods of numerical integration and simulation algorithms. … The authors have written a fundamental book on contemporary probability theory and its applications. The book can be strongly recommended to theorists and applied scientists.” (Jordan M. Stoyanov, zbMATH 1472.60002, 2021)”

• Department of Mathematics, University of Aarhus, Aarhus, Denmark

  Ole E. Barndorff-Nielsen

• Department of Mathematics, University of Oslo, Oslo, Norway

  Fred Espen Benth

• Department of Mathematics, Imperial College London, London, UK

  Almut E. D. Veraart

Ole Barndorff-Nielsen is well known for his manifold contributions to the theory and applications of probability and mathematical statistics, as described in the introductions of The Fascination of Probability, Statistics and Their Applications, Springer 2016. He has contributed to various fields, including statistical inference, sedimentology, infinite divisibility and Levy theory, homogeneous turbulence, and financial econometrics. Together with Jürgen Schmiegel he has founded the field of ambit stochastics.

Fred Espen Benth’s research focuses on stochastic analysis and its applications to energy and finance. He has contributed to risk management analysis of financial markets for weather and energy, as well as theoretical developments of stochastic calculus, including non-semimartingale stochastic integration. Recently he has developed stochastic volatility models and autoregressive processes in the infinite dimensional context.

Almut E. D. Veraart is a statistician and probabilist with an interest in developing stochastic models and statistical methods for finance, energy markets and weather and environmental variables. Her main methodological contributions are in statistical finance focusing on stochastic volatility modelling and estimation based on high-frequency data and in spatio-temporal statistics dealing with simulation and inference for ambit fields.

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