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Nondegenerate SDEs with jumps and their hypoelliptic properties. (English) Zbl 1283.60084

Summary: We study ‘nondegenerate’ SDEs with jumps. These include SDEs satisfying the ‘point-wise positive’ condition and the ones satisfying the (nonstationary) Hörmander’s condition. We show that solutions of these SDEs have hypoelliptic properties. Our result is based on the Malliavin calculus on the Wiener-Poisson space. In case of the continuous SDE, it extends and refines works based on the Malliavin calculus on the Wiener space.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J75 Jump processes (MSC2010)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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