On mean exit time from a curvilinear domain

Abstract

Abstract In this short communication, we consider a mean exit time problem for a non-degenerate, two-dimensional, coupled diffusion process M t = ( x t , y t ) in the interior of a curvilinear domain D ψ = { ( x , y ) ∈ R + 2 : y > ψ ( x ) } with a C 2 -boundary, where x t is any arbitrary diffusion process and y t 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.