On the last zero process of a spectrally negative Lévy process

Erik J. Baurdoux, J. M. Pedraza

Published 2020-03-15Version 1

Let $X$ be a spectrally negative L\'evy process and consider $g_t$ the last time $X$ is below the level zero before time $t\geq 0$. We derive an It\^o formula for the three dimensional process $\{(g_t,t,X_t), t\geq 0 \}$ and its infinitesimal generator using a perturbation method for L\'evy processes. We also find an explicit formula for calculating functionals that include the whole path of the length of current positive excursion at time $t\geq 0$,q $U_t:=t-g_t$. These results are applied to optimal prediction problems for the last zero $g:=\lim_{t \rightarrow \infty} g_t$, when $X$ drifts to infinity. Moreover, the joint Laplace transform of $(U_{e_q},X_{e_q})$ where $e_q$ is an independent exponential time is found and a formula for a density of the $q$-potential measure of the process $\{(U_t,X_t,),t\geq 0 \}$ is derived.