Smooth Densities of Stochastic Differential Equations Forced by Degenerate Stable Type Noises

Abstract

Malliavin calculus for stochastic differential equations (SDEs) forced by degenerate stable like noises has been intensively studied recently, see e.g. [Hao, Peng and Zhang, J. Theoret. Probab. 34 (2021); Zhang, Ann. Probab. 42(5) (2012); Zhang, Ann. Probab. 45 (2017)]. In this paper, we derive a simple inequality as a replacement of Norris type lemma and use it to show that two families of degenerate SDEs with stable like noises admit smooth density functions.