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The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE

Johannes Brutsche, Angelika Rohde

TL;DR

This work analyzes the MLE for the level parameter $\rho$ in oscillating Brownian motion under infill sampling. The log-likelihood is highly non-standard due to discontinuities in $\rho$, leading to a nine-regime decomposition and a stable Poisson-type limit for the MLE when $n\to\infty$. The main result shows $n$-consistency and a stable limit $\arg\sup_{z} \ell(z L_1^{\rho_0}(X))$, where the local time $L_1^{\rho_0}(X)$ scales both the drift and the Poisson intensity. A local-time estimator enables practical confidence intervals, making the inference robust to unobserved local time. The analysis leverages a semimartingale decomposition, stable convergence theory, and a tailored Jacod-type framework for discontinuous limits. This coupling of local time with Poisson-type limits represents a novel mechanism in high-frequency inference for state-dependent volatility changes.

Abstract

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $ρ_0$, we prove in the infill asymptotics that the MLE is $n$-consistent, where $n$ denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in $ρ_0$, the analysis is highly non-standard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument $ρ$ is to $ρ_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neigborhood of $ρ_0$ asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in $ρ$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $ρ_0$.

The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE

TL;DR

This work analyzes the MLE for the level parameter in oscillating Brownian motion under infill sampling. The log-likelihood is highly non-standard due to discontinuities in , leading to a nine-regime decomposition and a stable Poisson-type limit for the MLE when . The main result shows -consistency and a stable limit , where the local time scales both the drift and the Poisson intensity. A local-time estimator enables practical confidence intervals, making the inference robust to unobserved local time. The analysis leverages a semimartingale decomposition, stable convergence theory, and a tailored Jacod-type framework for discontinuous limits. This coupling of local time with Poisson-type limits represents a novel mechanism in high-frequency inference for state-dependent volatility changes.

Abstract

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality , we prove in the infill asymptotics that the MLE is -consistent, where denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in , the analysis is highly non-standard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument is to . Correspondingly, the MLE is successively excluded to lay outside a compact set, a -neighborhood and finally a -neigborhood of asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in , it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at .

Paper Structure

This paper contains 20 sections, 22 theorems, 387 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The log-likelihood function and its fundamental properties
  3. Proof of Proposition \ref{['prop:n-consistency']}
  4. The MLE is not outside a $1/\sqrt{n}$-neigborhood of $\rho_0$
  5. The MLE is not outside a $1/n$-neigborhood of $\rho_0$
  6. Proof of Proposition \ref{['prop:convergence_ell_n']}
  7. Preliminaries on stable convergence
  8. Proof of Proposition \ref{['prop:convergence_ell_n']}
  9. Proof of Proposition \ref{['prop:stable_fidi_process']}
  10. Proof of Theorem \ref{['thm:MLE_limiting_distribution']}
  11. Notation
  12. Preliminary results
  13. Local time estimator for OBM
  14. Expansion of the normalized log-likelihood function
  15. Proofs of the results of Section \ref{['Section:likelihood']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

On the extended probability space, where the right-hand side is a well-defined $\mathbb{R}$-valued random variable almost surely.

Figures (2)

  • Figure 1: Different realizations of the log-likelihood function, simulated for the parameters $\alpha=0.5$, $\beta=0.2$, $\rho_0=0$ and $n=1,000$ on a grid of the interval $[-1,1]$ with gridsize $10^{-6}$. (a) A path in the 'global' environment. (b) A path in the $1/\sqrt{n}$-environment. (c) A path in the $1/n$-environment of the true parameter. (d) An average over $50$ realizations of the likelihood in the $1/n$-environment is given.
  • Figure 2: Route of proof of the $n$-consistency.

Theorems & Definitions (52)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Proposition 2.4
  • Lemma 3.1
  • Lemma 3.2
  • ...and 42 more