Non-linear Shot Noise: Lévy, Noah, & Joseph

We introduce and study a generic non-linear Shot Noise system-model. Shots of random magnitudes arrive to the system stochastically, following an arbitrary time-homogeneous Poisson point process. After ‘hitting’ the system, the magnitude of an arriving shot decays to zero. The decay is governed by an arbitrary differential-equation dynamics. Shots are independent, and their overall effect on the system is additive: the systemʹs noise level at time t equals the sum of the magnitudes, at time t, of all the shots arriving to the system prior to time t. The resulting Shot Noise is: (i) a Lévy process when the decay-dynamics are degenerate; (ii) a Lévy-driven Ornstein–Uhlenbeck process when the decay-dynamics are linear; and, (iii) a stationary non-Markov process when the decay-dynamics are non-linear. The resulting Shot Noise admits an underlying Lévy structure—which we explicitly compute, and can yield both the Noah effect and the Joseph effect. Closed-form analytic formulae for various statistics are derived, including: the log-Laplace transform and cumulants of the stationary noise level; the process’ auto-covariance function; and, the process’ range-of-dependence.