Density behaviour related to Lévy processes

Abstract

Let p t ( x ) p_t(x) , f t ( x ) f_t(x) and q t ∗ ( x ) q_t^*(x) be the densities at time t t of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of p t ( x ) p_t(x) , f t ( x ) f_t(x) and q t ∗ ( x ) q_t^*(x) , when t t is small and x x is large. Then for large x x , these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if p t ( x ) p_t(x) is comparable to t ν ( x ) t\nu (x) , as t → 0 t\rightarrow 0 and x → ∞ x\rightarrow \infty , then so is f t ( x ) f_t(x) .