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LAD estimation of locally stable SDE

Oleksii M. Kulyk, Hiroki Masuda

Abstract

We prove the asymptotic mixed normality of the least absolute deviation (LAD) estimator for a locally $α$-stable stochastic differential equation (SDE) observed at high frequency, where $α\in(0,2)$. We investigate both ergodic and non-ergodic cases, where the terminal sampling time diverges or is fixed, respectively, under different sets of assumptions. The objective function for the LAD estimator is expressed in a fully explicit form without necessitating numerical integration, offering a significant computational advantage over the existing non-Gaussian stable quasi-likelihood approach.

LAD estimation of locally stable SDE

Abstract

We prove the asymptotic mixed normality of the least absolute deviation (LAD) estimator for a locally -stable stochastic differential equation (SDE) observed at high frequency, where . We investigate both ergodic and non-ergodic cases, where the terminal sampling time diverges or is fixed, respectively, under different sets of assumptions. The objective function for the LAD estimator is expressed in a fully explicit form without necessitating numerical integration, offering a significant computational advantage over the existing non-Gaussian stable quasi-likelihood approach.

Paper Structure

This paper contains 27 sections, 25 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. Conventions.
  3. Setup and assumptions
  4. Main results
  5. Finite observation horizon
  6. Infinite observation horizon
  7. Proofs, I: Consistency at sub-optimal rate
  8. Contrast function: definition and representation
  9. Estimates for martingale and predictable parts
  10. Basic consistency
  11. Improved consistency for $\alpha<1$
  12. Consistency at sub-optimal rate
  13. Proofs, II: the main statements
  14. Linearization
  15. Localization
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let Assumptions Ass1 to AssStab hold. Assume also the following identifiability conditions: Then, the LAD estimator satisfies

Theorems & Definitions (58)

  • Remark 2.1
  • Remark 2.2
  • Example 2.1: Euler scheme
  • Remark 2.3
  • Example 2.2: Improved Euler schemes
  • Example 2.3: Linear ODE
  • Example 2.4: Bernoulli's ODE
  • Remark 2.4
  • Remark 2.5
  • Example 2.5
  • ...and 48 more