Continuous-time security pricing. A utility gradient approach

We consider a (not necessarily complete) continuous-time security market with semimartingale prices and general information filtration. In such a setting, we show that the first-order conditions for optimality of an agent maximizing a 'smooth' (but not necessarily additive) utility can be formulated as the martingale property of prices, after normalization by a 'state-price' process. The latter is given explicitly in terms of the agent's utility gradient, which is in turn computed in closed form for a wide class of dynamic utilities, including stochastic differential utility, habit-forming utilities, and extensions.