In this paper, we investigate the Malliavin differentiability and density smoothness of solutions to stochastic differential equations (SDEs) with non-Lipschitz coefficients. Specifically, we consider equations of the form dXt = b(Xt)dt + σ (Xt)dWt, X0 = x0 where the drift b(·) and diffusion σ(·) may violate the global Lipschitz condition but satisfy weaker assumptions such as Hölder continuity, linear growth, and non-degeneracy. By employing Malliavin calculus theory, large deviation principles, and Fokker–Planck equations, we establish comprehensive results concerning the existence and uniqueness of solutions, their Malliavin differentiability, and the smoothness properties of density functions. Our main contributions include (1) proving the Malliavin differentiability of solutions under the standard linear growth condition combined with Hölder continuity; (2) establishing the existence and smoothness of density functions using Norris lemma and the Bismut–Elworthy–Li formula; and (3) providing optimal estimates for density functions through large deviation theory. These results have significant applications in financial mathematics (e.g., CIR, CEV, and Heston models), biological system modeling (e.g., stochastic population dynamics and neuronal and epidemiological models), and other scientific domains.
Qu et al. (Tue,) studied this question.