The shadow price of information in continuous time decision problems

We formulate a continuous time stochastic control problem and establish the existence of the shadow price of information. This shadow price is the Lagrange multiplier for the constraint that the control be adapted or predictable; it is a stochastic process of integrable variation, and, in one formulation, it is a martingale. The results are applied to problems of security investment, selling an asset, and economic growth. In the last application, it is shown that the existence of the shadow price of information implies the validity of the stochastic maximum principle