On the Estimations of Smooth Densities for Integro-differential Operators

Abstract

In this article, the conditions on the triple (a, b, c) of coefficients of the semi-group (P(t, x, dy)) _t≥0 with generator

are given such that (P(t, x, dy)) _t≥0 has a smooth density (p(t, x, y)) _t≥0, that is, P(t, x, dy) = p(t, x, y)dy. Where

and [Ktilde](x, A) = ∫_ℤ I _A∖{0}(c(x, z))dz is a Lévy kernel. And an estimation of p(t, x, y) is also given under some non-degeneracy assumptions.

Key Words:

• Estimation of smooth density
• Integro-differential operator
• Malliavin calculus
• Wiener–Poisson functional
• Stochastic differential equation with jumps

Acknowledgments

Research supported in part by the National Science Foundation of China (Project 19971072) and the Natural Science Foundation of Education Committee of Jiangsu Province of China (03KJB110135).