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Prediction-based inference for integrated diffusions with high-frequency data

Emil S. Jørgensen, Michael Sørensen

Abstract

We consider parametric inference for an ergodic and stationary diffusion process, when the data are high-frequency observations of the integral of the diffusion process. Such data are obtained via certain measurement devices, or if positions are recorded and speed is modelled by a diffusion. In finance, realized volatility or variations thereof can be used to construct observations of the latent integrated volatility process. Specifically, we assume that the integrated process is observed at equidistant, deterministic time points and consider the high-frequency/infinite horizon asymptotic scenario, where the number of observations, the sampling frequency and the time of the last observation all go to infinity. Subject to mild standard regularity conditions on the diffusion model, we prove the asymptotic existence and uniqueness of a consistent estimator for useful and tractable classes of prediction-based estimating functions. Asymptotic normality of the estimator is obtained under an additional assumption on the rates. The proofs are based on the useful Euler-Ito expansions of transformations of diffusions and integrated diffusions, which we study in some detail.

Prediction-based inference for integrated diffusions with high-frequency data

Abstract

We consider parametric inference for an ergodic and stationary diffusion process, when the data are high-frequency observations of the integral of the diffusion process. Such data are obtained via certain measurement devices, or if positions are recorded and speed is modelled by a diffusion. In finance, realized volatility or variations thereof can be used to construct observations of the latent integrated volatility process. Specifically, we assume that the integrated process is observed at equidistant, deterministic time points and consider the high-frequency/infinite horizon asymptotic scenario, where the number of observations, the sampling frequency and the time of the last observation all go to infinity. Subject to mild standard regularity conditions on the diffusion model, we prove the asymptotic existence and uniqueness of a consistent estimator for useful and tractable classes of prediction-based estimating functions. Asymptotic normality of the estimator is obtained under an additional assumption on the rates. The proofs are based on the useful Euler-Ito expansions of transformations of diffusions and integrated diffusions, which we study in some detail.
Paper Structure (19 sections, 10 theorems, 150 equations)

This paper contains 19 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and concepts
  4. Model assumption
  5. Prediction-based estimating functions
  6. Euler-Itô expansions
  7. Diffusion processes
  8. Integrated diffusions
  9. Limit theory for integrated diffusions
  10. Asymptotic theory
  11. Simple predictor spaces
  12. 1-lag predictor spaces
  13. Proofs and auxiliary results
  14. Auxiliary results
  15. Proofs
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $f \in \mathcal{C}^4_p(S)$. Then there exist $\mathcal{F}^n_i$-measurable random variables $\varepsilon_{1,i}$ and $\varepsilon_{2,i}$ such that where $\varepsilon_{1,i} \sim \mathcal{N}(0,1)$ and is independent of $\mathcal{F}^n_{i-1}$, and $\varepsilon_{2,i}$ satisfies the moment expansions

Theorems & Definitions (20)

  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 5.2
  • Lemma 5.3
  • Lemma 5.5
  • Theorem 5.6
  • Lemma 6.1
  • ...and 10 more