Existence of bounded solutions of fully nonlinear elliptic equations modelling the first passage time in cylindrical domains

The aim of this paper is to establish existence and uniqueness of bounded viscosity solutions of fully nonlinear second-order elliptic equations in domains like slabs and cylinders. A global Hölder regularity is also proved. The issue is motivated by different applications based on the First Passage Time (FPT). In particular, the equations modelize the expected value of the FPT for a time-homogeneous stochastic process. Explicit solutions have been computed for the Wiener process, and the in fluence of a drift term on the mean first hitting time has been also discussed.