On First-passage Problems for one-dimensional diffusions with random jump reflection at a boundary

Let $X(t)$ be a time-homogeneous one-dimensional diffusion process defined in $I \subset \rm I\!R,$ starting at $x \in I$ and let $ c\in I $ a barrier with $c <x.$ Suppose that, whenever the barrier $c$ is reached, the process $X$ is killed and continued as a new process $\widetilde X$ which makes a random jump from $c$ according to a given distribution, and then it starts again. First-passage-time problems for $\widetilde X$ are investigated, and closed form expressions are found for the expectation and the moment generating function of the first-passage time of $\widetilde X$ over a barrier $S >x.$ Moreover, the stationary distribution of $\widetilde X$ is studied, whenever it exists. Some explicit examples are also reported.