The aim of this paper is to study the value function of a fully convex Bolza problem with state constraints and under no coercivity assumptions. This requires that the state trajectories be of bounded variation rather than merely absolutely continuous. Our approach is based on the duality theory of classical convex analysis, and we establish a Fenchel-Young type equality between the value function of the Bolza problem and a suitable value function associated with its dual problem. The main result we present in this paper is a characteristic method that describes the evolution of the subgradients of the associated value functions. A few simple examples are provided to demonstrate the theory.