Computational Tools for the Shot Noise with Random Amplitude

Abstract

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.