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Abstract: Let $X$ be a L\-evy process with absolutely continuous L\-evy measure $ u$.Small time polynomial expansions of order $n$ in $t$ are obtained for the tails$PX {t}\geq{}y$ of the process, assuming smoothness conditions on the L\-evydensity away from the origin. By imposing additional regularity conditions onthe transition density $p {t}$ of $X {t}$, an explicit expression for theremainder of the approximation is also given. As a byproduct, polynomialexpansions of order $n$ in $t$ are derived for the transition densities of theprocess. The conditions imposed on $p {t}$ require that its derivatives remainuniformly bounded away from the origin, as $t\to{}0$; such conditions are shownto be satisfied for symmetric stable L\-evy processes as well as for otherrelated L\-evy processes of relevance in mathematical finance. The expansionsseem to correct asymptotics previously reported in the literature.



Autor: José E. Figueroa-López, Christian Houdré

Fuente: https://arxiv.org/







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