The connection between the maximum principle and dynamic programming in stochastic control

There are usually two ways to study optimal stochastic control problems: Pontryagin's maximum principle and Bellman's dynamic programming, involving an adjoint process ψ and the value function V, respectively. The classical result on the connection between the maximum principle and dynamic programming is known as ψ(t)=V x(t,◯(t)) where ◯(∣) is the optimal path. In this paper we establish a nonsmooth version of the classical result by employing the notions of super_ and sub_differential introduced by Crandall and Lions. Thus the illusory assumption that V is differentiate is dispensed with.