Statistical Inference and Malliavin Calculus

The derivative of the log-likelihood function, known as score function, plays a central role in parametric statistical inference. It can be used to study the asymptotic behavior of likelihood and pseudo-likelihood estimators. For instance, one can deduce the local asymptotic normality property which leads to various asymptotic properties of these estimators. In this article we apply Malliavin Calculus to obtain the score function as a conditional expectation. We then show, through different examples, how this idea can be useful for asymptotic inference of stochastic processes. In particular, we consider situations where there are jumps driving the data process.