  Stochastic modelling of fatigue (and other material's deterioration), as well
as of cumulative damage in risk theory, are often based on compound sums of
independent random variables, where the number of addends is represented by an
independent counting process. We consider here a cumulative model where,
instead of a renewal process (as in the Poisson case), a linear birth (or Yule)
process is used. This corresponds to the assumption that the frequency of
\textquotedblleft damage" increments accelerates according to the increasing
number of \textquotedblleft damages". We start from the partial differential
equation satisfied by its transition density, in the case of exponentially
distributed addends, and then we generalize it by introducing a
space-derivative of convolution type (i.e. defined in terms of the Laplace
exponent of a subordinator). Then we are concerned with the solution of
integro-differential equations, which, in particular cases, reduce to
fractional ones. Correspondingly, we analyze the related cumulative jump
processes under a general infinitely divisible distribution of the (positive)
jumps. Some special cases (such as the stable, tempered stable, gamma and
Poisson) are presented.
