Joint asymptotic distribution of certain path functionals of the reflected process

Abstract

Let τ(x) be the first time that the reflected process Y of a L´evy process X crosses x > 0. The main aim of this paper is to investigate the joint asymptotic distribution of Y (t) = X(t) − inf0≤s≤t X(s) and the path functionals Z(x) = Y (τ(x)) − x and m(t) = sup0≤s≤t Y (s) − y ∗(t), for a certain non-linear curve y ∗(t). We restrict to L´evy processes X satisfying Cram´er’s condition, a non-lattice condition and the moment conditions that E[|X(1)|] and E[exp(γX(1))|X(1)|] are finite (where γ denotes the Cram´er coefficient). We prove that Y (t) and Z(x) are asymptotically independent as min{t, x} → ∞ and characterise the law of the limit (Y∞, Z∞). Moreover, if y ∗(t) = γ−1 log(t) and min{t, x} → ∞ in such a way that t exp{−γx} → 0, then we show that Y (t), Z(x) and m(t) are asymptotically independent and derive the explicit form of the joint weak limit (Y∞, Z∞, m∞). The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law (Y∞, Z∞).