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Multifunction Estimation in a Time-Discretized Skorokhod Reflection Problem

Nicolas Marie

TL;DR

The paper addresses estimating an unknown time-varying convex body $C(t)$ that governs a time-discretized Skorokhod reflection of a diffusion, using a Moreau sweeping framework. It introduces a discrete approximation $\\overline{X}_{j+1}$ and defines a convex-hull estimator $\\widehat{C}_{N,j}$ from $N$ independent copies, enabling set-valued inference for diffusion-constraint dynamics. The authors prove $1$-D consistency with rate $N$ (and $N d_{\rm H}(\\widehat{C}_{N,j}, C(t_j)) \to 0$ when $\\sigma$ is constant$)$ and establish pointwise consistency in higher dimensions under a monotonicity assumption on $C$. This advances copy-based statistical inference for reflected diffusions by providing a concrete, consistent estimator for the evolving constraint set. The results have potential applications in fields like ethology where the target’s territory evolves over time and is not directly observed.

Abstract

This paper deals with a consistent estimator of the multifunction involved in a time-discretized Skorokhod reflection problem defined by a stochastic differential equation and a Moreau sweeping process.

Multifunction Estimation in a Time-Discretized Skorokhod Reflection Problem

TL;DR

The paper addresses estimating an unknown time-varying convex body that governs a time-discretized Skorokhod reflection of a diffusion, using a Moreau sweeping framework. It introduces a discrete approximation and defines a convex-hull estimator from independent copies, enabling set-valued inference for diffusion-constraint dynamics. The authors prove -D consistency with rate (and when is constant and establish pointwise consistency in higher dimensions under a monotonicity assumption on . This advances copy-based statistical inference for reflected diffusions by providing a concrete, consistent estimator for the evolving constraint set. The results have potential applications in fields like ethology where the target’s territory evolves over time and is not directly observed.

Abstract

This paper deals with a consistent estimator of the multifunction involved in a time-discretized Skorokhod reflection problem defined by a stochastic differential equation and a Moreau sweeping process.
Paper Structure (7 sections, 4 theorems, 46 equations)

This paper contains 7 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs
  4. Proof of Theorem \ref{['convergence_unidimensional']}
  5. Proof of Lemma \ref{['lemma_convergence_unidimensional']}
  6. Proof of Theorem \ref{['convergence_multidimensional']}
  7. Proof of Lemma \ref{['lemma_convergence_multidimensional']}

Key Result

Theorem 2.1

Assume that there exists a constant $\mathfrak m_{\tt C} > 0$ such that, for every $t\in [0,T]$ and $x\in {\tt C}(t)$, $|x|\leqslant\mathfrak m_{\tt C}$. For any $j\in J_n\backslash\{0\}$, and if in addition $\sigma$ is constant, then

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma A.1
  • Lemma A.2