The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If X(t) is a one-dimensional diffusion with jumps, starting from a random position η ∈ [a, b], let be τ a,b the time at which X(t) first 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; it can be seen as a problem of optimization.