Asymptotic inference for a stochastic differential equation with uniformly distributed time delay

TL;DR

This paper investigates the asymptotic properties of likelihood functions for affine stochastic differential equations with uniformly distributed delays, providing insights into estimator behavior under various parameter conditions.

Contribution

It establishes the local asymptotic properties of likelihood functions for such equations, including normality and quadraticity, and explores the implications for maximum likelihood estimators.

Findings

Proves local asymptotic normality for certain parameter values

Demonstrates local asymptotic mixed normality in other cases

Analyzes the asymptotic behavior of maximum likelihood estimators

Abstract

For affine stochastic differential equation with uniformly distributed time delay the local asymptotic properties of the likelihood function are studied. Local asymptotic normality, local asymptotic mixed normality, periodic local asymptotic mixed normality or local asymptotic quadraticity is proved for different values of the parameter. Applications to the asymptotic behaviour of the maximum likelihood estimator of the parameter based on continuous sample are given.