In stochastic control the value function is rarely smooth. Thus one needs to define extended solutions of the Hamilton-Jacobi equation (H-J). The author introduces a new notion of second order generalized derivative which invovles Brownian motion. He proves an Itô formula for functions \(f(t,x)\) which are continuously differentiable in \(x\) with Lipschitz derivative and are Lipschitz continuous in \(t\). He then defines a generalized solution of the second order H-J and shows that the value function is in fact a generalized solution. Finally, without recourse to any control problem, he shows that any generalized solution of H-J is a viscosity subsolution and a viscosity solution is a generalized solution.