Shot Noise with Random Parameters

We give the description of the following model: $$U_{n}=X_{n}(Y_{n}+U_{n-1})$$ for $n>1$ in the case where the $X_{n}$ are i.d.d. random variables with probability density: $$A x^{A-1} , x \in [0,1] ,$$ $A$ is also a random variable distributed according to a Gamma law. The $Y_{n}$ are or deterministic and equal to $1$ or independent Gamma random variables. We use this model to compute the shot noise with random parameters. Keywords: Random difference equations, Shot noise, Volterra functions, differential-difference equations, Monte-carlo simulation.