Density behaviour related to Lévy processes

2019/12/09 by Chaumont, Loïc, Małecki, Jacek
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1912.04193

Abstract

Let p_t(x), f_t(x) and q_t^*(x) be the densities at time 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), f_t(x) and q_t^*(x), when t is small and x is large. Then for large 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) is comparable to tν(x), as t→0 and x→∞, then so is f_t(x).

Related

• The asymptotic behavior of the density of the supremum of L 'evy\n processes
• Estimates of densities for Lévy processes with lower intensity of large jumps
• Intrinsic small time estimates for distribution densities of Lévy processes
• Compound kernel estimates for the transition probability density of a Lévy process in \rn
• Small-time expansions for the transition distributions of Lévy processes