By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\fracλ{n}\,\|φ\|_{\infty} \leq \|{\rm Hess}\ φ\|_{\infty}\leq c(D)λ\,\|φ\|_{\infty}$ holds for any Dirichlet eigenfunction $φ$ of $-Δ$ with eigenvalue $λ$. Our results provide the sharp Hessian estimate $\|{\rm Hess}\ φ\|_{\infty}\lesssim λ^{\frac{n+3}{4}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.