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

Abundo Mario

Control and Cybernetics

2022

Vol. 51, No. 1

31--42

We report some additional examples of explicit solu- tions 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 2 [a, b], let be τa,b the time at which X(t) first exits the interval (a, b), and πa = P(X(τa,b) ≤ a) the probability of exit from the left of (a, b). Given a probability q 2 (0, 1), the problem consists in finding the density g of η (if it exists) such that πa = q; it can be seen as a problem of optimization.