On mean exit time from a curvilinear domain

In this short communication, we consider a mean exit time problem for a non-degenerate, two-dimensional, coupled diffusion process Mt=(xt,yt) in the interior of a curvilinear domain Dψ={(x,y)∈R+2:y>ψ(x)} with a C2-boundary, where xt is any arbitrary diffusion process and yt is a geometric Brownian motion evolving under non-explosive conditions, and ψ(.) is a real-valued, positive, increasing, continuous function such that ψ(0)≥0. It is proved that, under certain conditions, the mean exit time is a logarithmic function associated with a certain second-order nonlinear ordinary differential equation. At the end of the note, we shall present several examples to illustrate our main result.