We consider a jump-diffusion process that appears to be suitable to describe the displacement performed during the rising phase by a single myosin head along an actin filament as reported in numerous current publications by T. Yanagida and his co-workers. We obtain the transition density of the considered process under the Markovian assumption that the instants of occurrence of the jumps are distributed according to a Poisson process. Two different distributions for the random amplitude of the jumps are considered in detail: the Gaussian distribution and a mixture of two Gaussian distributions. By means of the present phenomenological model, certain significant analogies and qualitative agreements with the biological data are pointed out.