Small-time expansions for the transition distributions of Lévy processes

By José E. Figueroa-López, C. Houdr'e. 2008

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Let X=(Xt)t>=0 be a Levy process with absolutely continuous Levy measure [nu]. Small-time expansions of arbitrary polynomial order in t are obtained for the tails , y>0, of the process, assuming smoothness conditions on the Levy density away from the origin. By imposing additional regularity conditions on the transition density pt of Xt, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order n in t are derived for the transition densities of the process. The conditions imposed on pt require that, away from the origin, its derivatives remain uniformly bounded as t-->0. Such conditions are then shown to be satisfied for symmetric stable Levy processes as well as some tempered stable Levy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4433cfb8b8aba6a4aecc01ced884487bfa7a431