Smooth density for the Solution of Scalar SDEs with Locally Lipschitz Coefficients under Hörmander Condition

Abstract: In this paper the existence of a smooth density is proved for the solution of an SDE, with locally Lipschitz coefficients and semi-monotone drift, under H\"ormander condition. We prove the nondegeneracy condition for the solution of the SDE, from it an integration by parts formula would result in the Wiener space. To this end we construct a sequence of SDEs with globally Lipschitz coefficients whose solutions converge to the original one and use some Lyapunov functions to show the uniformly boundedness of the p-moments of the solutions and their Malliavin derivatives.