Differentiability of the value function in stochastic models

Résumé

Most of the recent contributions to intertemporal optimization models of consumption and capital accumulation aim at the development of a theory of intertemporal resource allocation under uncertainty. One of the standard tools for solving such models is Stochastic Dynamic Programming.A disadvantage of the application of this technique is that although solutions are shown to exist and are easily computable it has not been proved that, under standard assumptions, the policy function is differentiable or, equivalently, that the value function is twice differentiable.This feature is of particular interest in the qualitative analysis of optimal solutions.This paper shows that if the objective function is C, a-concave and stochastic shock s satisfy standard assumptions, then the value function is C.