First-exit times for Poisson shot noise

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 T_b at which a given level b > 0 is exceeded. An integral equation for the joint density of T_b and X(T_b) 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 ψ_b(θ) = ∫₀^∞ e-^θt P(T_b > t) dt of the function t → P(T_b > 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(T_b > t, X(t) < u), that is the joint distribution function of sup_0≤s≤t X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).