Computational Tools for the Shot Noise with Random Amplitude

The following random recurrency: $$W_{n+1} = U_{n+1} ( W_n + \Lambda_{n+1} )$$ where $W_{-1}=0.$ is known to be associated with the shot noise : $$W_{t} = \sum_{0<t_k<t} \Lambda_{k} e^{-(t-t_k)}$$ where the $t_k$ are the dates of a Poisson process, the $U_i ;\ i = 0,1,...$ are independent uniform variables. This can be extended to the triggered shot noise : $$W_{t} = \sum_{0<t_{lk}<t} \Lambda_{lk} e^{-(t-t_{lk})} , l=1,2,3,...$$ The random amplitudes $\Lambda_i,\ i = 0,1,...$ are positive Dufresne independent variables or also even density variables which characteristic function given by generalized hypergeometric function. Keywords: Random Difference Equations,Triggered Shot Noise, Iterated Cosine-Bessel functions, Logarithmic Distribution Functions,Saddle Point Method, Asymptotic Calculations, Generalized Hypergeometric Functions, Psi Functions.