Idée - Stochastic Processes - # Ambit Stochastics and Stochastic Volatility Modeling

Advances in Ambit Stochastics: Honoring the Legacy of Ole E. Barndorff-Nielsen

Q: How can the insights from ambit stochastics be applied to other areas of science and engineering beyond turbulence modeling?

Ambit stochastics, with its robust framework for modeling random phenomena in space-time, can be effectively applied to various fields beyond turbulence modeling. For instance, in environmental science, ambit fields can be utilized to model spatially and temporally varying phenomena such as pollution dispersion, where the random fluctuations in pollutant concentration can be captured using the stochastic volatility inherent in ambit processes. In finance, the concept of stochastic volatility, a key aspect of ambit stochastics, can enhance models for asset prices, particularly in capturing the irregularities and jumps observed in financial markets. The flexibility of ambit fields allows for the incorporation of various market conditions and external shocks, leading to more accurate pricing and risk assessment. Moreover, in the field of telecommunications, ambit stochastics can be employed to model signal propagation in complex environments, where the random nature of the medium can significantly affect signal quality. The ability to model these random fluctuations in a coherent framework can lead to improved designs for communication systems. In epidemiology, ambit fields can be used to model the spread of diseases over time and space, accounting for the stochastic nature of transmission rates and recovery processes. This can provide valuable insights into the dynamics of disease outbreaks and inform public health interventions.

Q: What are the potential limitations or challenges in extending the statistical inference techniques developed for ambit processes to more general classes of random fields?

While the statistical inference techniques developed for ambit processes are powerful, several limitations and challenges arise when attempting to extend these methods to more general classes of random fields. One significant challenge is the complexity of the underlying structure of general random fields, which may not possess the same properties as ambit fields, such as the explicit definition of an ambit set or the presence of a L´evy basis. Another limitation is the identifiability of models. In ambit stochastics, the separation of deterministic kernel functions from stochastic volatility components aids in model identification. However, in more general random fields, the lack of such clear separability can complicate the estimation of parameters and hinder the development of robust inference techniques. Additionally, the assumptions required for the statistical inference methods, such as continuity and integrability conditions, may not hold in more general settings. This can lead to difficulties in establishing convergence results or deriving limit theorems, which are crucial for statistical inference. Finally, computational challenges may arise when dealing with more complex random fields, as the analytical tractability that ambit stochastics offers may diminish. This can necessitate the development of new numerical methods or approximation techniques, which can be resource-intensive and may not always yield satisfactory results.

Q: How might the connections between ambit fields and stochastic partial differential equations inspire new approaches to modeling and analyzing complex spatio-temporal phenomena?

The connections between ambit fields and stochastic partial differential equations (SPDEs) provide a rich framework for modeling and analyzing complex spatio-temporal phenomena. By viewing ambit fields as solutions to SPDEs, researchers can leverage the mathematical tools and techniques developed for SPDEs to gain insights into the behavior of ambit processes. This relationship allows for the incorporation of spatial and temporal dynamics into the modeling process, enabling a more comprehensive understanding of phenomena such as fluid dynamics, heat transfer, and wave propagation. The flexibility of ambit fields in capturing random fluctuations can enhance the modeling of systems where traditional deterministic approaches fall short. Moreover, the ability to express ambit fields in terms of integral operators and Green's functions opens avenues for developing new numerical methods for solving SPDEs. This can lead to more efficient simulations of complex systems, facilitating the exploration of parameter spaces and the assessment of model robustness. Additionally, the insights gained from studying the interplay between ambit fields and SPDEs can inspire new theoretical developments, such as the establishment of limit theorems for realized variance in spatio-temporal contexts. This can enhance the statistical inference techniques available for analyzing data from complex systems, leading to improved predictions and decision-making. In summary, the synergy between ambit fields and SPDEs not only enriches the theoretical landscape but also provides practical tools for addressing real-world challenges in various scientific and engineering domains.

Concepts de base

This article surveys key developments in ambit stochastics, a research field pioneered by Ole E. Barndorff-Nielsen, and highlights emerging trends in this area. It focuses on statistical inference for stochastic volatility and the connections between ambit fields and stochastic partial differential equations.

Résumé

The article begins by introducing the concept of ambit fields, which provide a flexible yet analytically tractable class of random field models. It emphasizes Ole Barndorff-Nielsen's contributions to the foundation and advancement of ambit stochastics, particularly his work on stochastic volatility and the role of metatimes.

The article then reviews statistical inference for stochastic volatility in three key settings: Brownian semistationary (BSS) processes in the semimartingale and non-semimartingale cases, as well as ambit processes in a spatio-temporal context. For each case, the article presents the corresponding limit theorems for realized variance and related power variation measures, highlighting the importance of appropriate scaling to achieve convergence.

Next, the article discusses the connections between ambit fields and stochastic partial differential equations (SPDEs), showing how ambit fields can be expressed as mild solutions of parabolic SPDEs. It then reviews recent advances in volatility estimation for infinite-dimensional settings, including the development of semigroup-adjusted realized covariation and multipower variation measures.

The article concludes by highlighting the applications of ambit fields, particularly in the context of turbulence modeling, and provides an outlook on the latest developments in ambit stochastics.

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arxiv.org

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Citations

"Ambit fields provide a flexible, yet analytically tractable class of random field models, which go far beyond the widely used Gaussian random fields."
"Ole always stressed the importance of modelling stochastic volatility/intermittency via stochastic time change or, more generally, the concept of metatimes."
"The key component of an ambit field takes the form Y(x,t) = ∫_A(x,t) g(x,t;ξ,s)σ(ξ,s)L_T(dξ,ds)."

Idées clés tirées de

Research Frontiers in Ambit Stochastics: In memory of Ole E. Barndorff-Nielsen

by Fred Espen B... à arxiv.org 10-02-2024

https://arxiv.org/pdf/2410.00566.pdf

Research Frontiers in Ambit Stochastics: In memory of Ole E. Barndorff-Nielsen

Questions plus approfondies

How can the insights from ambit stochastics be applied to other areas of science and engineering beyond turbulence modeling?

Ambit stochastics, with its robust framework for modeling random phenomena in space-time, can be effectively applied to various fields beyond turbulence modeling. For instance, in environmental science, ambit fields can be utilized to model spatially and temporally varying phenomena such as pollution dispersion, where the random fluctuations in pollutant concentration can be captured using the stochastic volatility inherent in ambit processes.
In finance, the concept of stochastic volatility, a key aspect of ambit stochastics, can enhance models for asset prices, particularly in capturing the irregularities and jumps observed in financial markets. The flexibility of ambit fields allows for the incorporation of various market conditions and external shocks, leading to more accurate pricing and risk assessment.
Moreover, in the field of telecommunications, ambit stochastics can be employed to model signal propagation in complex environments, where the random nature of the medium can significantly affect signal quality. The ability to model these random fluctuations in a coherent framework can lead to improved designs for communication systems.
In epidemiology, ambit fields can be used to model the spread of diseases over time and space, accounting for the stochastic nature of transmission rates and recovery processes. This can provide valuable insights into the dynamics of disease outbreaks and inform public health interventions.

What are the potential limitations or challenges in extending the statistical inference techniques developed for ambit processes to more general classes of random fields?

While the statistical inference techniques developed for ambit processes are powerful, several limitations and challenges arise when attempting to extend these methods to more general classes of random fields. One significant challenge is the complexity of the underlying structure of general random fields, which may not possess the same properties as ambit fields, such as the explicit definition of an ambit set or the presence of a L´evy basis.
Another limitation is the identifiability of models. In ambit stochastics, the separation of deterministic kernel functions from stochastic volatility components aids in model identification. However, in more general random fields, the lack of such clear separability can complicate the estimation of parameters and hinder the development of robust inference techniques.
Additionally, the assumptions required for the statistical inference methods, such as continuity and integrability conditions, may not hold in more general settings. This can lead to difficulties in establishing convergence results or deriving limit theorems, which are crucial for statistical inference.
Finally, computational challenges may arise when dealing with more complex random fields, as the analytical tractability that ambit stochastics offers may diminish. This can necessitate the development of new numerical methods or approximation techniques, which can be resource-intensive and may not always yield satisfactory results.

How might the connections between ambit fields and stochastic partial differential equations inspire new approaches to modeling and analyzing complex spatio-temporal phenomena?

The connections between ambit fields and stochastic partial differential equations (SPDEs) provide a rich framework for modeling and analyzing complex spatio-temporal phenomena. By viewing ambit fields as solutions to SPDEs, researchers can leverage the mathematical tools and techniques developed for SPDEs to gain insights into the behavior of ambit processes.
This relationship allows for the incorporation of spatial and temporal dynamics into the modeling process, enabling a more comprehensive understanding of phenomena such as fluid dynamics, heat transfer, and wave propagation. The flexibility of ambit fields in capturing random fluctuations can enhance the modeling of systems where traditional deterministic approaches fall short.
Moreover, the ability to express ambit fields in terms of integral operators and Green's functions opens avenues for developing new numerical methods for solving SPDEs. This can lead to more efficient simulations of complex systems, facilitating the exploration of parameter spaces and the assessment of model robustness.
Additionally, the insights gained from studying the interplay between ambit fields and SPDEs can inspire new theoretical developments, such as the establishment of limit theorems for realized variance in spatio-temporal contexts. This can enhance the statistical inference techniques available for analyzing data from complex systems, leading to improved predictions and decision-making.
In summary, the synergy between ambit fields and SPDEs not only enriches the theoretical landscape but also provides practical tools for addressing real-world challenges in various scientific and engineering domains.