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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.

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)
Full Text: DOI
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