Bootstrap method for misspecified ergodic L\'{e}vy driven stochastic differential equation models

TL;DR

This paper develops a bootstrap method to accurately approximate the asymptotic distribution of estimators in misspecified Lévy-driven stochastic differential equation models, regardless of the noise distribution.

Contribution

It introduces a bootstrap approach that remains valid under model misspecification and does not require exact knowledge of the noise distribution.

Findings

Bootstrap method accurately approximates asymptotic distribution

Method works for both Gaussian and non-Gaussian noise

Theoretical validation provided for both correctly specified and misspecified models

Abstract

In this paper, we consider possibly misspecified stochastic differential equation models driven by L\'{e}vy processes. Regardless of whether the driving noise is Gaussian or not, Gaussian quasi-likelihood estimator can estimate unknown parameters in the drift and scale coefficients. However, in the misspecified case, the asymptotic distribution of the estimator varies by the correction of the misspecification bias, and consistent estimators for the asymptotic variance proposed in the correctly specified case may lose theoretical validity. As one of its solutions, we propose a bootstrap method for approximating the asymptotic distribution. We show that our bootstrap method theoretically works in both correctly specified case and misspecified case without assuming the precise distribution of the driving noise.