insight - Scientific Computing - # Hawkes Processes

Functional Limit Theorems for Subcritical and Critical Hawkes Processes: A Comprehensive Analysis

Q: Could there be alternative stochastic processes or models that might be more suitable than Hawkes processes for capturing the dynamics of systems with long-range dependencies but without the specific self-exciting structure?

While Hawkes processes are effective for modeling systems with both self-excitation and long-range dependencies, alternative stochastic processes and models might be more suitable when the self-exciting structure is absent. Here are a few alternatives: Fractional Brownian Motion (fBm): fBm is a generalization of Brownian motion that incorporates long-range dependence through its Hurst exponent. It's suitable for modeling time series with long memory but without a clear self-exciting mechanism. Autoregressive Fractionally Integrated Moving Average (ARFIMA) models: ARFIMA models combine the features of autoregressive (AR), integrated (I), and moving average (MA) models with fractional differencing to capture long memory. They are widely used in time series analysis when both short- and long-range dependencies are present. Fractional Lévy Processes: These processes generalize Lévy processes by incorporating long-range dependence. They are suitable for modeling phenomena with jumps and long memory, such as financial returns or internet traffic. Agent-Based Models (ABMs): ABMs simulate the behavior of individual agents in a system and their interactions. By incorporating heterogeneity and feedback loops, ABMs can generate long-range dependencies without explicitly assuming self-excitation. They are particularly useful for studying the emergent behavior of complex systems. The choice of the most suitable model depends on the specific characteristics of the system being studied, the available data, and the research question being addressed.

Q: How does the concept of "memory" in strongly critical Hawkes processes, as evidenced by long-range dependencies, relate to the notion of memory in other complex systems, such as biological systems or social networks?

The concept of "memory" in strongly critical Hawkes processes, as evidenced by long-range dependencies, shares intriguing similarities with the notion of memory in other complex systems: Biological Systems: In biological systems, memory can manifest as the persistence of past events' influence on present behavior. For instance, the immune system "remembers" past encounters with pathogens, leading to a faster and more efficient response upon re-exposure. Similarly, in strongly critical Hawkes processes, the heavy-tailed kernel function allows past events to exert a long-lasting influence on future event occurrences, leading to long-range dependencies. Social Networks: Social networks exhibit memory through the persistence of information and influence. For example, a viral post on social media can continue to generate engagement and influence opinions long after its initial posting. This persistence of influence mirrors the long-range dependencies observed in strongly critical Hawkes processes, where past events continue to shape the probability of future events. Common Underlying Mechanisms: The presence of "memory" in these diverse systems suggests common underlying mechanisms. In strongly critical Hawkes processes, the heavy-tailed kernel function acts as a mechanism for propagating the influence of past events. Similarly, in biological and social systems, feedback loops, network effects, and information cascades can serve as mechanisms for perpetuating the impact of past events. Implications for Understanding Complexity: Recognizing the commonality of "memory" across different complex systems can provide insights into their behavior and dynamics. By studying the mechanisms underlying long-range dependencies in Hawkes processes, we may gain a deeper understanding of how memory shapes the behavior of other complex systems, leading to more accurate models and predictions.

Core Concepts

The long-run behavior of Hawkes processes is determined by the average number and dispersion of child events, with subcritical processes exhibiting distinct behaviors from critical processes, particularly regarding long-range dependencies.

Abstract

Bibliographic Information: Horst, U., & Xu, W. (2024). Functional Limit Theorems for Hawkes Processes. arXiv preprint arXiv:2401.11495v2.
Research Objective: This paper aims to analyze the asymptotic behavior of subcritical and critical Hawkes processes by establishing functional limit theorems, focusing on the impact of child event dispersion on long-term dynamics.
Methodology: The authors utilize a duality method to derive an exponential-aﬃne representation of the Fourier-Laplace functional for Hawkes processes. They then employ this representation to analyze the asymptotic behavior of the processes under different conditions on the kernel function, particularly focusing on the dispersion of child events.
Key Findings:
- The moment condition requiring the integral of the square root of time multiplied by the kernel function to be finite is not sharp for establishing the functional central limit theorem (FCLT) for subcritical Hawkes processes.
- The long-term behavior of a subcritical Hawkes process is determined by the function representing the dispersion of child events. Different asymptotic behaviors, including classical FCLT, non-classical FCLT, and degenerate FCLT, are observed depending on the limit of this function.
- Critical Hawkes processes exhibit long-range dependencies, and their long-run behavior is dictated by the dispersion coefficient. Weakly critical processes, with a finite dispersion coefficient, are not asymptotically stationary and converge to a CIR model without mean-reversion. Strongly critical processes, with an infinite dispersion coefficient, display long-range dependencies but satisfy classical FLLNs and FCLTs.
Main Conclusions: The paper provides a comprehensive analysis of the long-run behavior of both subcritical and critical Hawkes processes, demonstrating the crucial role of child event dispersion in determining their asymptotic properties. The results have implications for understanding the dynamics of various phenomena modeled by Hawkes processes, particularly in fields like finance and seismology.
Significance: This research significantly contributes to the theoretical understanding of Hawkes processes, particularly by establishing functional limit theorems under weaker conditions and providing a nuanced analysis of critical processes. The findings have implications for modeling and analyzing systems with self-exciting behavior, particularly in scenarios where long-range dependencies are crucial.
Limitations and Future Research: The paper primarily focuses on Hawkes processes with constant exogenous intensity. Exploring the asymptotic behavior under more general intensity functions could be a potential avenue for future research. Additionally, investigating the implications of the findings for specific applications of Hawkes processes, such as financial modeling or earthquake prediction, would be valuable.

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Functional Limit Theorems for Hawkes Processes

by Ulrich Horst... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2401.11495.pdf

Functional Limit Theorems for Hawkes Processes

Deeper Inquiries

How can the insights from this research be applied to improve the modeling of financial markets, particularly in capturing the dynamics of high-frequency trading or volatility clustering?

This research provides valuable insights into the dynamics of systems with self-exciting and long-range dependency properties, which are frequently observed in financial markets, particularly in high-frequency trading and volatility clustering. Here's how the findings can be applied:

Improved Modeling of High-Frequency Trading: High-frequency trading is characterized by the rapid arrival of orders and trades, often exhibiting self-excitation, where past events trigger future ones. This research's focus on the impact of the kernel function, particularly its tail behavior, provides a framework for selecting and calibrating Hawkes processes to accurately capture the dynamics of order flow, trade clustering, and price impact. For instance, understanding the dispersion of child events can help in developing more realistic market simulators and optimizing order execution strategies.

Capturing Volatility Clustering: Volatility clustering, the tendency of large price fluctuations to cluster together, is a well-documented stylized fact of financial markets. The research's findings on critical Hawkes processes, particularly those exhibiting long-range dependencies, offer a promising avenue for modeling volatility. By linking the tail behavior of the kernel function to the observed long memory in volatility, one can develop more accurate volatility forecasting models and risk management tools.

Calibrating Realistic Models: The explicit exponential-affine representation of the Fourier-Laplace functional provided in the research facilitates the calibration of Hawkes process models to empirical data. This is crucial for practical applications in finance, where accurate parameter estimation is essential for reliable forecasting and decision-making.

Understanding Market Microstructure: The research's insights into the interplay between the kernel function and the long-run behavior of Hawkes processes can shed light on the underlying market microstructure mechanisms driving self-excitation and long-range dependencies. This deeper understanding can contribute to the development of more effective regulatory policies and market design.

Could there be alternative stochastic processes or models that might be more suitable than Hawkes processes for capturing the dynamics of systems with long-range dependencies but without the specific self-exciting structure?

While Hawkes processes are effective for modeling systems with both self-excitation and long-range dependencies, alternative stochastic processes and models might be more suitable when the self-exciting structure is absent. Here are a few alternatives:

Fractional Brownian Motion (fBm): fBm is a generalization of Brownian motion that incorporates long-range dependence through its Hurst exponent. It's suitable for modeling time series with long memory but without a clear self-exciting mechanism.

Autoregressive Fractionally Integrated Moving Average (ARFIMA) models: ARFIMA models combine the features of autoregressive (AR), integrated (I), and moving average (MA) models with fractional differencing to capture long memory. They are widely used in time series analysis when both short- and long-range dependencies are present.

Fractional Lévy Processes: These processes generalize Lévy processes by incorporating long-range dependence. They are suitable for modeling phenomena with jumps and long memory, such as financial returns or internet traffic.

Agent-Based Models (ABMs): ABMs simulate the behavior of individual agents in a system and their interactions. By incorporating heterogeneity and feedback loops, ABMs can generate long-range dependencies without explicitly assuming self-excitation. They are particularly useful for studying the emergent behavior of complex systems.
The choice of the most suitable model depends on the specific characteristics of the system being studied, the available data, and the research question being addressed.

How does the concept of "memory" in strongly critical Hawkes processes, as evidenced by long-range dependencies, relate to the notion of memory in other complex systems, such as biological systems or social networks?

The concept of "memory" in strongly critical Hawkes processes, as evidenced by long-range dependencies, shares intriguing similarities with the notion of memory in other complex systems:

Biological Systems: In biological systems, memory can manifest as the persistence of past events' influence on present behavior. For instance, the immune system "remembers" past encounters with pathogens, leading to a faster and more efficient response upon re-exposure. Similarly, in strongly critical Hawkes processes, the heavy-tailed kernel function allows past events to exert a long-lasting influence on future event occurrences, leading to long-range dependencies.

Social Networks: Social networks exhibit memory through the persistence of information and influence. For example, a viral post on social media can continue to generate engagement and influence opinions long after its initial posting. This persistence of influence mirrors the long-range dependencies observed in strongly critical Hawkes processes, where past events continue to shape the probability of future events.

Common Underlying Mechanisms: The presence of "memory" in these diverse systems suggests common underlying mechanisms. In strongly critical Hawkes processes, the heavy-tailed kernel function acts as a mechanism for propagating the influence of past events. Similarly, in biological and social systems, feedback loops, network effects, and information cascades can serve as mechanisms for perpetuating the impact of past events.

Implications for Understanding Complexity: Recognizing the commonality of "memory" across different complex systems can provide insights into their behavior and dynamics. By studying the mechanisms underlying long-range dependencies in Hawkes processes, we may gain a deeper understanding of how memory shapes the behavior of other complex systems, leading to more accurate models and predictions.