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

TL;DR

This paper investigates the local asymptotic properties of the growth rate in a jump-type Cox-Ingersoll-Ross process driven by Brownian motion and a subordinator, covering ergodic and non-ergodic cases with advanced mathematical techniques.

Contribution

It establishes the local asymptotic normality, quadraticity, and mixed normality of the growth rate in different regimes, using Malliavin calculus and jump analysis.

Findings

LAN in the subcritical case

LAQ in the critical case

LAMN in the supercritical case

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.