Generalized Solutions of the Hamilton–Jacobi Equation of Stochastic Control

Abstract

A second-order derivative based on Brownian motion is introduced. Using this derivative, an Ito-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 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.