Joint distributions of the infimum, the supremum and the end of Brownian motion with jumps

In the paper, we compute the joint distributions of the infimum, the supremum and the end of the Brownian motion with jumps. The final time of the Brownian motion with jumps is taken at the exponentially distributed random time independent of the process. This corresponds to the Laplace transform with respect to the deterministic time of the joint distributions under consideration.