An inverse problem for the first-passage place of some diffusion processes with random starting point. Issue 6 (1st November 2020) - Abundo, Mario
Abstract: We study an inverse problem for the first-passage place of a one-dimensional diffusion process X ( t ) (also with jumps), starting from a random position η ∈ [ a, b ] . Let be τ a, b the first time at which X ( t ) exits the interval ( a, b ), and π a = P ( X ( τ a, b ) ≤ a ) the probability of exit from the left of ( a, b ) . Given a probability q ∈ ( 0, 1 ), the problem consists in finding the density g of η (if it exists) such that π a = q . Some explicit examples are reported.