Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An unbounded intensity model for point processes

Kim Christensen, Alexei Kolokolov

TL;DR

The paper develops a point-process model with locally unbounded intensity, capturing intensity bursts that concentrate events around a stopping time while keeping the compensator finite to avoid explosion. It introduces a heavy-traffic/infill framework to enable inference over a fixed time horizon and proposes a nonparametric, online test statistic based on the local intensity difference, complemented by an observed-variance estimator. A change-of-frequency approach is provided to distinguish bursts from jumps, and Monte Carlo simulations show size control and high power, even under self-exciting baselines. The empirical application to EUR/USD high-frequency data reveals frequent bursts linked to higher volatility and illiquidity, with drift bursts more likely when the order book is fragile and order flow is imbalanced. The results offer a practical toolkit for real-time screening of abnormal trading activity and deepen understanding of the microstructure consequences of intensity bursts.

Abstract

We develop a model for point processes on the real line, where the intensity can be locally unbounded without inducing an explosion. In contrast to an orderly point process, for which the probability of observing more than one event over a short time interval is negligible, the bursting intensity causes an extreme clustering of events around the singularity. We propose a nonparametric approach to detect such bursts in the intensity. It relies on a heavy traffic condition, which admits inference for point processes over a finite time interval. With Monte Carlo evidence, we show that our testing procedure exhibits size control under the null, whereas it has high rejection rates under the alternative. We implement our approach on high-frequency data for the EUR/USD spot exchange rate, where the test statistic captures abnormal surges in trading activity. We detect a nontrivial amount of intensity bursts in these data and describe their basic properties. Trading activity during an intensity burst is positively related to volatility, illiquidity, and the probability of observing a drift burst. The latter effect is reinforced if the order flow is imbalanced or the price elasticity of the limit order book is large.

An unbounded intensity model for point processes

TL;DR

The paper develops a point-process model with locally unbounded intensity, capturing intensity bursts that concentrate events around a stopping time while keeping the compensator finite to avoid explosion. It introduces a heavy-traffic/infill framework to enable inference over a fixed time horizon and proposes a nonparametric, online test statistic based on the local intensity difference, complemented by an observed-variance estimator. A change-of-frequency approach is provided to distinguish bursts from jumps, and Monte Carlo simulations show size control and high power, even under self-exciting baselines. The empirical application to EUR/USD high-frequency data reveals frequent bursts linked to higher volatility and illiquidity, with drift bursts more likely when the order book is fragile and order flow is imbalanced. The results offer a practical toolkit for real-time screening of abnormal trading activity and deepen understanding of the microstructure consequences of intensity bursts.

Abstract

We develop a model for point processes on the real line, where the intensity can be locally unbounded without inducing an explosion. In contrast to an orderly point process, for which the probability of observing more than one event over a short time interval is negligible, the bursting intensity causes an extreme clustering of events around the singularity. We propose a nonparametric approach to detect such bursts in the intensity. It relies on a heavy traffic condition, which admits inference for point processes over a finite time interval. With Monte Carlo evidence, we show that our testing procedure exhibits size control under the null, whereas it has high rejection rates under the alternative. We implement our approach on high-frequency data for the EUR/USD spot exchange rate, where the test statistic captures abnormal surges in trading activity. We detect a nontrivial amount of intensity bursts in these data and describe their basic properties. Trading activity during an intensity burst is positively related to volatility, illiquidity, and the probability of observing a drift burst. The latter effect is reinforced if the order flow is imbalanced or the price elasticity of the limit order book is large.
Paper Structure (21 sections, 10 theorems, 220 equations, 10 figures, 1 table)

This paper contains 21 sections, 10 theorems, 220 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Intensity burst of a point process
  3. Identification
  4. Test statistic
  5. Observed asymptotic local variance
  6. How to separate a burst from a jump?
  7. Simulation study
  8. Empirical application
  9. A burst or jump?
  10. State of the market near an intensity burst
  11. What is the relationship with a drift burst?
  12. Conclusion
  13. Proofs
  14. Proof of Lemma \ref{['lemma:consistency']}
  15. Proof of Theorem \ref{['theorem:clt']}
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose that Assumption assumption:rate holds. Then, under $\mathscr{H}_{0}$, as $n \rightarrow \infty$ and $\delta_{n} \rightarrow 0$ such that $n \delta_{n} \rightarrow \infty$, it holds for all fixed $t \in [0,T]$ that Moreover, under $\mathscr{H}_{1}$,

Figures (10)

  • Figure 1: Intensity burst in the EUR/USD with a drift burst.
  • Figure 2: Example of a simulated intensity burst.
  • Figure 3: Daily price and log-return in EUR/USD.
  • Figure 4: Trading activity in EUR/USD.
  • Figure 5: Candidate times for an intensity burst.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Lemma 2
  • Theorem 6
  • ...and 1 more