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Deep learning of point processes for modeling high-frequency data

Yoshihiro Gyotoku, Ioane Muni Toke, Nakahiro Yoshida

TL;DR

This work develops a theoretically grounded framework for applying deep learning to high-frequency point-process data with covariates. By casting intensity learning as estimation of functions in a flexible class ${\mathfrak F_T}$ (including deep nets) and establishing an oracle-type inequality, the authors derive convergence rates for the prediction error under ${\alpha}$-mixing covariates and a compatibility condition. They extend the modeling to marked point processes via the marked ratio model and demonstrate both one-step and two-step estimation schemes, with simulations showing the two-step approach often yields superior predictive performance. An empirical application to limit-order-book data and high-frequency trades corroborates the practical utility of multiplicative, covariate-aware intensity models learned by neural networks. Overall, the paper provides a rigorous path from nonparametric learning theory to actionable, covariate-rich intensity models for financial data, including concrete rate results and robust estimation strategies.

Abstract

We investigate applications of deep neural networks to a point process having an intensity with mixing covariates processes as input. Our generic model includes Cox-type models and marked point processes as well as multivariate point processes. An oracle inequality and a rate of convergence are derived for the prediction error. A simulation study shows that the marked point process can be superior to the simple multivariate model in prediction. We apply the marked ratio model to real limit order book data

Deep learning of point processes for modeling high-frequency data

TL;DR

This work develops a theoretically grounded framework for applying deep learning to high-frequency point-process data with covariates. By casting intensity learning as estimation of functions in a flexible class (including deep nets) and establishing an oracle-type inequality, the authors derive convergence rates for the prediction error under -mixing covariates and a compatibility condition. They extend the modeling to marked point processes via the marked ratio model and demonstrate both one-step and two-step estimation schemes, with simulations showing the two-step approach often yields superior predictive performance. An empirical application to limit-order-book data and high-frequency trades corroborates the practical utility of multiplicative, covariate-aware intensity models learned by neural networks. Overall, the paper provides a rigorous path from nonparametric learning theory to actionable, covariate-rich intensity models for financial data, including concrete rate results and robust estimation strategies.

Abstract

We investigate applications of deep neural networks to a point process having an intensity with mixing covariates processes as input. Our generic model includes Cox-type models and marked point processes as well as multivariate point processes. An oracle inequality and a rate of convergence are derived for the prediction error. A simulation study shows that the marked point process can be superior to the simple multivariate model in prediction. We apply the marked ratio model to real limit order book data

Paper Structure

This paper contains 21 sections, 7 theorems, 147 equations, 10 figures, 1 table.

Table of Contents

  1. Point process with covariates and deep learning
  2. Rate of convergence of the error
  3. Application to deep learning
  4. The marked ratio model
  5. A simulation study
  6. Description of the numerical example
  7. One-step ratio estimation
  8. Two-step ratio estimation
  9. Fitting results
  10. Comparison of the methods and convergence results
  11. Robustness with respect to shapes of the neural networks
  12. Parsimony and computational time
  13. An application to high-frequency trades and LOB data
  14. Some basic estimates and proof of Theorem \ref{['0607170208']}
  15. Preparations
  16. ...and 6 more sections

Key Result

Theorem 2.4

Let $\xi$ be any positive number. Then there exists a constant ${C_{0}}$ depending on ${\color{black}\gamma}$, ${\tt h}$, $\||\lambda^*|\|_\infty$, ${\sf d}_N$, ${C_{*}}$ and $\xi$, such that whenever ${\tt T}\geq2\vee\{\xi(\log{\tt T})^2\log{\cal N}_T\}$ and ${\cal N}_T\geq2$. Here ${\cal N}_T={\cal N}_{T,\delta}$ is the covering number of ${\mathfrak F}_T$ by the $\delta$-balls with respect to

Figures (10)

  • Figure 1: Simulation study --- Estimated functions $\hat{l}_1^{i,k_i}(x,y)$ by the one-step estimation method. True functions are plotted as dotted lines of the color of the corresponding estimated function.
  • Figure 2: Simulation study --- Estimated functions $\hat{l}_2^{i}(x)$ by the two-step estimation method. True functions are plotted as dotted lines of the color of the corresponding estimated function.
  • Figure 3: Simulation study --- Estimated functions $\hat{l}_2^{i,k_i}(y)$ by the two-step estimation method. True functions are plotted as dotted lines of the color of the corresponding estimated function.
  • Figure 4: Simulation study --- Estimated probabilities $\hat{p}_1^{i,k_i}(x,y)$ by the one-step estimation method. True functions are plotted as dotted lines of the color of the corresponding estimated function.
  • Figure 5: Simulation study --- Estimated probabilities $\hat{p}_2^{i,k_i}(x,y)$ by the two-step estimation method. True functions are plotted as dotted lines of the color of the corresponding estimated function.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Remark 3.2
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 5.4
  • ...and 1 more