A NEW FORMULA FOR SOME LINEAR STOCHASTIC EQUATIONS WITH APPLICATIONS

We give a representation of the solution for a stochastic linear equation of the form X-t = Y-t + integral((0,t]) X-s-dZ(s) where Z is a cadlag semimartingale and Y is a cadlag 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 Levy processes, the resulting X is called a generalized Ornstein-Uhlenbeck process.