There has been a growing interest in constructing stationary measures with known multifractal properties. In an earlier paper, the authors introduced the multifractal products of stochastic processes (MPSP) and provided basic properties concerning convergence, nondegeneracy, and scaling of moments. This paper considers a subclass of MPSP which is determined by jump processes with i.i.d. exponentially distributed interjump times. Particularly, the information dimension and a multifractal spectrum of the MPSP are computed. As a side result it is shown that the random partitions imprinted naturally by a family of Poisson point processes are sufficient to determine the spectrum in this case.