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Local asymptotic properties for the growth rate of a jump-type CIR process

Mohamed Ben Alaya, Ahmed Kebaier, Gyula Pap, Ngoc Khue Tran

TL;DR

The paper addresses the problem of estimating the growth rate parameter $b$ in a one-dimensional jump-type CIR process driven by a Brownian motion and a subordinator, under both continuous and high-frequency discrete observations. It develops a unified local asymptotic framework, proving LAN in the subcritical regime, LAQ in the critical regime, and LAMN in the supercritical regime, with explicit limiting objects and rates for each setting. The methodology blends explicit Radon–Nikodym derivatives, Malliavin calculus for the score and its representation via Skorohod integrals, and a refined jump analysis to handle infinite-activity subordinators, yielding precise log-likelihood expansions and negligible remainder control. These results extend classical diffusion-based LAN/LAQ/LAMN theory to jump-type affine processes, providing a foundation for efficient inference on growth-rate parameters in financial and interest-rate models under high-frequency data.

Abstract

In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To obtain these results, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.

Local asymptotic properties for the growth rate of a jump-type CIR process

TL;DR

The paper addresses the problem of estimating the growth rate parameter in a one-dimensional jump-type CIR process driven by a Brownian motion and a subordinator, under both continuous and high-frequency discrete observations. It develops a unified local asymptotic framework, proving LAN in the subcritical regime, LAQ in the critical regime, and LAMN in the supercritical regime, with explicit limiting objects and rates for each setting. The methodology blends explicit Radon–Nikodym derivatives, Malliavin calculus for the score and its representation via Skorohod integrals, and a refined jump analysis to handle infinite-activity subordinators, yielding precise log-likelihood expansions and negligible remainder control. These results extend classical diffusion-based LAN/LAQ/LAMN theory to jump-type affine processes, providing a foundation for efficient inference on growth-rate parameters in financial and interest-rate models under high-frequency data.

Abstract

In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To obtain these results, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.

Paper Structure

This paper contains 30 sections, 35 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Continuous observations
  4. Subcritical case
  5. Critical case
  6. Supercritical case
  7. Discrete observations
  8. Subcritical case
  9. Critical case
  10. Supercritical case
  11. Proof of main results
  12. Proof of Theorem \ref{['Thm_subcritical']}
  13. Proof of Theorem \ref{['Thm_critical']}
  14. Proof of Theorem \ref{['Thm_supercritical']}
  15. Proof of Theorems \ref{['theorem1']}, \ref{['theorem2']} and \ref{['theorem3']}
  16. ...and 15 more sections

Key Result

Proposition 1.1

BBKP17 For all $a \in \mathbb{R}_+$, $b \in \mathbb{R}$, $\sigma \in \mathbb{R}_{++}$, $y_0 \in \mathbb{R}_+$ and Lévy measure $m$ on $\mathbb{R}_{++}$ satisfying condition (A1), there is a pathwise unique strong solution $Y^b=(Y_t^b)_{t\in\mathbb{R}_+}$ of the SDE eq1 such that $\widehat{\mathrm{P} and immigration mechanism Furthermore, the infinitesimal generator of $Y^b$ takes the form where

Theorems & Definitions (60)

  • Proposition 1.1
  • Lemma 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Theorem 2.8
  • ...and 50 more