Weak Solutions of a Partial Differential Equation of Dynamic Programming

Abstract

A formal dynamic programming argument relates the “value function of a stochastic optimal control problem” with the solution of a nonlinear parabolic partial differential equation. In cases in which the partial differential equation is degenerate, it may not have a classical solution but may have a weak solution in the sense of the theory of Schwartz distributions. It has been an open question as to whether a weak solution of the partial differential equation does equal the value function of the stochastic optimal control problem. This paper shows that, roughly, whenever an associated uncontrolled system has an appropriately behaved density function, the equality holds. It is also shown that an optimal control law may be determined by minimizing a Hamiltonian formed in terms of the partial derivatives of the weak solution of the partial differential equation.