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

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 Lp distance (p > 1) with g, the last time X is negative. The solution is substantially more difficult compared to the case 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 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 non-increasing and non-negative 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 non-linear 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.