Shot Noise with Random Parameters

Abstract

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.