The quadratic variation of Brownian motion is used to give a new definition of a generalized Hessian for nonsmooth functions, totally independently of any gradients. A calculus is developed for the generalized Hessian of functions whose gradient is Lipschitz. It is also shown that for convex functions the Hessian is nonempty.
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U. G. Haussmann (1992) studied this question.
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