Lp optimal prediction of the last zero of a spectrally negative Lévy process

Abstract

Given a spectrally negative Lévy process X drifting to infinity, (inspired on the early ideas of Shiryaev (2002)) we are interested in finding a stopping time that minimises the ${\mathit{L}}^{\mathit{p}}$ distance ($\mathit{p}>1$) with g, the last time X is negative. The solution is substantially more difficult compared to the case $\mathit{p}=1$, for which it was shown by Baurdoux and Pedraza (2020) that it is optimal to stop as soon as X exceeds a constant barrier. In the case of $\mathit{p}>1$ treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process that incorporates the length of the current positive excursion away from 0. We show that an optimal stopping time is now given by the first time that X exceeds a nonincreasing and nonnegative curve depending on the length of the current positive excursion away from 0. We further characterise the optimal boundary and the value function as the unique solution of a nonlinear system of integral equations within a subclass of functions. As examples, the case of a Brownian motion with drift and a Brownian motion with drift perturbed by a Poisson process with exponential jumps are considered.