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Estimation of parameters and local times in a discretely observed threshold diffusion model

Sara Mazzonetto, Paolo Pigato

Abstract

We consider a simple mean reverting diffusion process, with piecewise constant drift and diffusion coefficients, discontinuous at a fixed threshold. We discuss estimation of drift and diffusion parameters from discrete observations of the process, with a generalized moment estimator and a maximum likelihood estimator. We develop the asymptotic theory of the estimators when the time horizon of the observations goes to infinity, considering both cases of a fixed time lag (low frequency) and a vanishing time lag (high frequency) between consecutive observations. In the setting of low frequency observations and infinite time horizon we also study the convergence of three local time estimators, that are already known to converge to the local time in the setting of high frequency observations and fixed time horizon. We find that these estimators can behave differently, depending on the assumptions on the time lag between observations.

Estimation of parameters and local times in a discretely observed threshold diffusion model

Abstract

We consider a simple mean reverting diffusion process, with piecewise constant drift and diffusion coefficients, discontinuous at a fixed threshold. We discuss estimation of drift and diffusion parameters from discrete observations of the process, with a generalized moment estimator and a maximum likelihood estimator. We develop the asymptotic theory of the estimators when the time horizon of the observations goes to infinity, considering both cases of a fixed time lag (low frequency) and a vanishing time lag (high frequency) between consecutive observations. In the setting of low frequency observations and infinite time horizon we also study the convergence of three local time estimators, that are already known to converge to the local time in the setting of high frequency observations and fixed time horizon. We find that these estimators can behave differently, depending on the assumptions on the time lag between observations.
Paper Structure (23 sections, 14 theorems, 123 equations, 2 figures, 1 table)

This paper contains 23 sections, 14 theorems, 123 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work.
  3. Outline.
  4. Estimators and convergence results
  5. Drift parameters inference
  6. Volatility parameters inference
  7. Local time estimation
  8. Derivation of the estimators and comments on the results
  9. Derivation of the estimators.
  10. On the equivalence of drift GME and dMLE, beyond the ergodic regime.
  11. Drift estimation in high frequency and infinite horizon.
  12. Multi-threshold.
  13. Numerical studies
  14. Comparison of drift estimators
  15. The asymptotic normality constant in the drift estimator
  16. ...and 8 more sections

Key Result

Theorem 1

Let $h>0$ fixed. Then These results hold also substituting estimator $\overline{b}^\pm$ with $\widehat{b}^\pm$, but in this case the convergence in $i)$ holds in probability (weak consistency).

Figures (2)

  • Figure 1: We compare the GME $\overline{b}^\pm_{h,N}$ and the dMLE $\hat{b}^\pm_{h,N}$, with $h=1$ fixed, varying the time horizon (length of the time series) $N$. We plot in this figure the mean squared error of each estimator, on $10^{3}$ simulated time series, as a function of the length of the time series, in a log-log plot in base $10$.
  • Figure 2: CLT in Theorem \ref{['th:fixed:h']}-ii), with parameters as in Table \ref{['table:simulation_parameters']}. We plot the density of the theoretical distribution of the estimation error, with variance estimated on $10^{3}$ simulated paths, and compare it with the distribution of the rescaled error on $n=10^3$ paths, and $N=10^3,10^4,10^5$ observations on each path.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 15 more