Malliavin calculus approach to statistical inference for Levy driven SDE's

TL;DR

This paper develops a Malliavin calculus framework to derive integral representations of likelihood functions for Levy-driven SDEs, enabling analysis of statistical properties and efficiency bounds.

Contribution

It introduces a novel Malliavin calculus approach to obtain likelihood representations for SDEs driven by tempered -stable processes, advancing statistical inference methods.

Findings

Derived integral representations for likelihood and its derivative.

Proved regularity of the statistical experiment.

Established Crame9r-Rao lower bounds for estimators.

Abstract

By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX_t=a_\theta(X_t)dt + dZ_t with a tempered \alpha-stable process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.