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.
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Some problems of first-crossing times over two time-dependent boundaries for one-dimensional jump-diffusion processes are considered. The moments of the first-crossing times over each boundary are shown to be the solutions of certain partial differential-difference equations with suitable outer conditions. An approach based on the Laplace transform allows us to compare the moments of the first-crossing times of the jump-diffusion process with those of the corresponding simple-diffusion without jumps. For some examples where the boundaries are constant, the results are illustrated graphically.
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