Application of the lent particle method to Poisson-driven SDEs

• 成果类型: Article
• 署名作者: Bouleau, Nicolas; Denis, Laurent
• 署名单位: Universite Paris Saclay; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0303-x
• 发表日期: 2011
• 页码: 403-433
• 关键词: absolute continuity; Levy processes; REGULARITY; density; CONSTRUCTION; calculus

摘要

We apply the Dirichlet form theory to stochastic differential equations with jumps as extension of Malliavin calculus reasoning. As in the continuous case, this weakens significantly the assumptions on the coefficients of the SDE. Thanks to the flexibility of the Dirichlet forms language, this approach brings also an important simplification which was neither available nor visible previously: an explicit formula giving the carr, du champ matrix, i.e., the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples. In this paper which uses the construction done in Bouleau and Denis (J. Funct. Analysis 257:1144-1174, 2009) we restrict ourselves to the existence of densities; smoothness will be studied separately.