Open Access
2014 Malliavin matrix of degenerate SDE and gradient estimate
Zhao Dong, Xuhui Peng
Author Affiliations +
Electron. J. Probab. 19: 1-26 (2014). DOI: 10.1214/EJP.v19-3120


In this article, we prove that the inverse of Malliavin matrix belongs to $L^p(\Omega,\mathbb{P})$ for a class of degenerate stochastic differential equation (SDE). The conditions required are similar to Hörmander's bracket condition, but we don't need 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.


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Zhao Dong. Xuhui Peng. "Malliavin matrix of degenerate SDE and gradient estimate." Electron. J. Probab. 19 1 - 26, 2014.


Accepted: 15 August 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1310.60076
MathSciNet: MR3256873
Digital Object Identifier: 10.1214/EJP.v19-3120

Primary: 60H10
Secondary: 60H07

Keywords: Degenerate stochastic differential equation , Gradient estimate , H\"{o}rmander condition , Malliavin calculus , Strong Feller

Vol.19 • 2014
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