Gradient estimates for SDEs without monotonicity type conditions

We prove gradient estimates for transition Markov semigroups (Pt) associated to SDEs driven by multiplicative Brownian noise having possibly unbounded C1-coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for DxPtφ of the formt|DxPtφ(x)|≤c(1+|x|k)‖φ‖∞,t∈(0,1], φ∈Cb(Rd), x∈Rd. We use two main tools. First, we consider a Feynman-Kac semigroup with potential V related to the growth of the coefficients and of their derivatives for which we can use a Bismut-Elworthy-Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift b having sublinear growth together with its derivative such that uniform estimates for DxPtφ without weights do not hold.