Malliavin matrix of degenerate SDE and gradient estimate. (English) Zbl 1310.60076

Dong, Zhao; Peng, Xuhui

Electron. J. Probab. 19, Paper No. 73, 26 p. (2014).

Summary: In this article, we prove that the inverse of the Malliavin matrix belongs to \(L^p(\Omega,\mathbb{P})\) for a class of degenerate stochastic differential equations (SDE). The conditions required are similar to Hörmander’s bracket condition, but we do not need that all coefficients of the SDE are smooth. Furthermore, we obtain a locally uniform estimate for the Malliavin matrix and a gradient estimate. We also prove that the semigroup generated by the SDE is strong Feller. These results are illustrated through examples.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07	Stochastic calculus of variations and the Malliavin calculus

Keywords:

degenerate stochastic differential equations; Malliavin matrix; gradient estimate; strong Feller semigroup; Hörmander’s bracket condition