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Asymptotic analysis of estimators of ergodic stochastic differential equations

Arnab Ganguly

Abstract

The paper studies asymptotic properties of estimators of multidimensional stochastic differential equations driven by Brownian motions from high-frequency discrete data. Consistency and central limit properties of a class of estimators of the diffusion parameter and an approximate maximum likelihood estimator of the drift parameter based on a discretized likelihood function have been established in a suitable scaling regime involving the time-gap between the observations and the overall time span. Our framework is more general than that typically considered in the literature and, thus, has the potential to be applicable to a wider range of stochastic models.

Asymptotic analysis of estimators of ergodic stochastic differential equations

Abstract

The paper studies asymptotic properties of estimators of multidimensional stochastic differential equations driven by Brownian motions from high-frequency discrete data. Consistency and central limit properties of a class of estimators of the diffusion parameter and an approximate maximum likelihood estimator of the drift parameter based on a discretized likelihood function have been established in a suitable scaling regime involving the time-gap between the observations and the overall time span. Our framework is more general than that typically considered in the literature and, thus, has the potential to be applicable to a wider range of stochastic models.

Paper Structure

This paper contains 13 sections, 22 theorems, 164 equations.

Table of Contents

  1. Introduction.
  2. Mathematical Framework
  3. Estimation of the drift parameter through approximate likelihood function
  4. Estimation of the diffusion parameter
  5. Assumptions
  6. Limit theorems for estimators
  7. General result on consistency
  8. Consistency and CLT of drift estimator
  9. Consistency and CLT of diffusion estimator
  10. A scaling regime and equivalent formulation of the results
  11. Auxiliary Results
  12. Auxiliary Lemma on $\psi$ and $\phi$
  13. Proofs of consistency and central limit theorem

Key Result

Theorem 3.1

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and $\{\mathcal{L}^\varepsilon, \varepsilon>0\}$ a family of stochastic processes (random fields) with path space, $C(\mathbb A, \mathbb R)$, where $\mathbb A \subset \mathbb R^{d_0}$ is measurable. Assume the following conditions hold. Let $\{\hat{\alpha}^\varepsilon\}$ be a family of random variables such that for each $\varepsilon>

Theorems & Definitions (50)

  • Remark 2.2: Existence of stationary distribution
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.7
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3: Consistency of the approximate maximum likelihood estimator
  • Remark 3.4
  • Theorem 3.5: CLT for estimator of the drift parameter
  • ...and 40 more