$L_p$ optimal prediction of the last zero of a spectrally negative Lévy process

Erik J. Baurdoux, J. M. Pedraza

Published 2020-03-15Version 1

Given a spectrally negative L\'evy process $X$ drifting to infinity, we are interested in finding a stopping time which minimises the $L^p$ distance ($p>1$) with $g$, the last time $X$ is negative. The solution is substantially more difficult compared to the $p=1$ case for which it was shown in \cite{baurdoux2018predicting} 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 which incorporates the length of the current 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 excursion away from $0$. We also show that smooth fit holds at the boundary.