A randomized first-passage problem for drifted Brownian motion subject to hold and jump from a boundary

We study an inverse first-passage-time problem for Wiener process $X(t)$ subject to hold and jump from a boundary $c\mathrm{.}$ Let be given a threshold $S>X(0) \ge c,$andadistributionfunction$F$on$[0, + \infty ).$ The problem consists in finding the distribution of the holding time at $c$ and the distribution of jumps from $c,$ so that the first-passage time of $X(t)$ through $S$ has distribution $F\mathrm{.}$