Exponential bounds for the density of the law of the solution of a SDE with locally Lipschitz coefficients

Under the uniform Hörmander’s hypothesis we study smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfy a monotonicity condition. To avoid non-integrability problems we use results about Malliavin differentiability based on the concepts of Ray Absolute Continuity and Stochastic Gateâux differentiability.