First-exit times for Poisson shot noise

Abstract

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time Tb at which a given level b > 0 is exceeded. An integral equation for the joint density of Tb and X(Tb) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT ψb(θ) = ∫0∞ e-θt P(Tb > t) dt of the function t → P(Tb > t) in terms of certain LTs derived from the distribution function H(x;t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(Tb > t, X(t) < u), that is the joint distribution function of sup0≤s≤t X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u). Copyright © 2001 by Marcel Dekker, Inc.