We give a representation of the solution for a stochastic linear equation of
the form Xt​=Yt​+∫(0,t]​Xs−​dZs​ where Z is a
c\'adl\'ag semimartingale and Y is a c\'adl\'ag adapted process with bounded
variation on finite intervals. As an application we study the case where Y
and −Z are nondecreasing, jointly have stationary increments and the jumps of
−Z are bounded by 1. Special cases of this process are shot-noise processes,
growth collapse (additive increase, multiplicative decrease) processes and
clearing processes. When Y and Z are, in addition, independent L\'evy
processes, the resulting X is called a generalized Ornstein-Uhlenbeck
process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP637 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org