The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem
type: Item
instance of: scholarly article
title: The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem
zbMATH Open document ID: 1009.60046
author: Atsushi Takeuchi
published in: Osaka Journal of Mathematics
publication date: 2003-04-07T00:00:00Z
review text: Let \(x_t\in{\mathbf R}^m\) be the solution of a stochastic differential equation driven by a Brownian motion and an \(\alpha\)-stable rotationally invariant Lévy process. Let \(\pi\) be a smooth mapping from \({\mathbf R}^m\) into \({\mathbf R}^n\) for \(m\geq n\). It is proved that under some conditions on the coefficients of the equation, the law of \(\pi(x_T)\) has a smooth density; this is called a partial hypoellipticity property. Conditions on the coefficients are technical, but in the continuous case, they reduce to classical assumptions and the result was proved by Taniguchi. The general result relies on a Malliavin calculus on the space of càdlàg paths which was previously worked out by the author and Komatsu.
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 1836034
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: stochastic differential equations with jumps
zbMATH Keywords: partial hypoellipticity
reviewed by: Jean Picard
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1836034