Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Estimation of a pure-jump stable Cox-Ingersoll-Ross process

Elise Bayraktar, Emmanuelle Clément

Abstract

We consider a pure-jump stable Cox-Ingersoll-Ross ($α$-stable CIR) process driven by a non-symmetric stable L{é}vy process with jump activity $α$ $\in$ (1, 2) and we address the joint estimation of drift, scaling and jump activity parameters from high-frequency observations of the process on a fixed time period. We first prove the existence of a consistent, rate optimal and asymptotically conditionally gaussian estimator based on an approximation of the likelihood function. Moreover, uniqueness of the drift estimators is established assuming that the scaling coefficient and the jump activity are known or consistently estimated. Next we propose easy-toimplement preliminary estimators of all parameters and we improve them by a one-step procedure.

Estimation of a pure-jump stable Cox-Ingersoll-Ross process

Abstract

We consider a pure-jump stable Cox-Ingersoll-Ross (-stable CIR) process driven by a non-symmetric stable L{é}vy process with jump activity (1, 2) and we address the joint estimation of drift, scaling and jump activity parameters from high-frequency observations of the process on a fixed time period. We first prove the existence of a consistent, rate optimal and asymptotically conditionally gaussian estimator based on an approximation of the likelihood function. Moreover, uniqueness of the drift estimators is established assuming that the scaling coefficient and the jump activity are known or consistently estimated. Next we propose easy-toimplement preliminary estimators of all parameters and we improve them by a one-step procedure.
Paper Structure (15 sections, 10 theorems, 218 equations)

This paper contains 15 sections, 10 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimating functions and limit theorems
  4. Main results
  5. Joint estimation
  6. Preliminary estimators and one-step improvement
  7. Proofs
  8. Proof of Proposition \ref{['P:moments']}
  9. Proof of Proposition \ref{['P:intstoch']}
  10. Estimation of $z^n$
  11. Proof of Theorem \ref{['Th : LFGN']}
  12. Proof of Theorem \ref{['Th : TCL']}
  13. Proof of Theorem \ref{['Th-local']}
  14. Proof of Theorem \ref{['Th-global']}
  15. Proof of Theorem \ref{['Estimation delta']}

Key Result

Proposition 2.1

Let $(X_t)_{t\in [0,1]}$ be the solution of eq:EDS. Then we have

Theorems & Definitions (15)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.1
  • Theorem 4.2
  • Remark 4.2
  • Theorem 4.3
  • Corollary 4.1
  • ...and 5 more