Estimation of stopping times for stopped self-similar random processes

Abstract

Let X=(X_t)_{tge 0} be a known process and T an unknown random time independent of X. Our goal is to derive the distribution of T based on an iid sample of X_T. Belomestny and Schoenmakers (Stoch Process Appl 126(7):2092–2122, 2015) propose a solution based the Mellin transform in case where X is a Brownian motion. Applying their technique we construct a non-parametric estimator for the density of T for a self-similar one-dimensional process X. We calculate the minimax convergence rate of our estimator in some examples with a particular focus on Bessel processes where we also show asymptotic normality.