We present sharp Hessian estimates of the form {Mathematical expression} for the solution of the viscous Hamilton-Jacobi equation {Mathematical expression}The smallest possible positive function g(t) is explicitly given in terms of the semiconvexity and semiconcavity parameters of V and S0, respectively. The optimal g does not depend on the viscosity parameter {Mathematical expression} . The potential V and the initial function S0 are allowed to grow quadratically