Generalized Solutions of the Hamilton–Jacobi Equation of Stochastic Control

A second-order generalized derivative based on Brownian motion is introduced. Using this derivative, an Itô–type formula is derived for functions $f(t,x)$, which are continuously differentiable in x with Lipschitz derivative and are Lipschitz continuous in t. It is then shown that the value function of a stochastic control problem is a “generalized” solution of a second-order Hamilton–Jacobi equation. Such solutions are analogous to the Clarke generalized solutions of first-order Hamilton–Jacobi equations. Finally, it is shown that any “generalized” solution is a viscosity subsolution and a viscosity solution is a “generalized” solution.