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Integration by parts formula and applications to equations with jumps. (English) Zbl 1243.60045
The authors are interested in the smoothness of the law at time $$t$$ of the solution $$X_t(\omega)\in\mathbb{R}^d$$ of some stochastic differential equation driven by a Poisson point measure having a portion-depending intensity. As the usual Malliavin calculus does not apply directly to such framework, they first develop a finite-dimensional version of conditional Malliavin calculus, which they can then use to derive smoothness results for the law of $$X_t$$.
The modification of the usual Malliavin method the authors introduce and implement aims at by-passing the impossibility of bounding the Ornstein-Uhlenbeck operator in their case. To proceed, and in order to estimate a Fourier transform $$\mathbb{E}(e^{i\xi F})$$, they use a convenient approaching sequence $$F_n$$ (going to $$F$$ in $$L^1$$) such that $|\mathbb{E}(e^{i\xi F})|\leq |\xi|\times \mathbb{E}[|F- F_n|]+ |\xi|^{-|\beta|}\times \mathbb{E}[|H_\beta(F_n)]= O(|\xi|^{-p}),$ where roughly $$H_\beta(F_n)$$ is given by the Malliavin inverse covariance matrix. Of course, a lost of estimates are necessary and performed, to let the machinery work.

##### MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 60G51 Processes with independent increments; Lévy processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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