Intertemporal asset pricing and the marginal utility of wealth

We consider the general class of discrete-time, finite-horizon intertemporal asset pricing models in which preferences for consumption at the intermediate dates are allowed to be state-dependent, satiated, non-convex and discontinuous, and the information structure is not required to be generated by a Markov process of state variables. We supply a generalized definition of marginal utility of wealth based on the Fréchet differential of the value operator that maps time t wealth into maximum conditional remaining utility. We show that in this general case all state-price densities/stochastic discount factors are fully characterized by the marginal utility of wealth of optimizing agents even if their preferences for intermediate consumption are highly irregular. Our result requires only the strict monotonicity of preferences for terminal wealth and the existence of a portfolio with positive and bounded gross returns. We also relate our generalized notion of marginal utility of wealth to the equivalent martingale measures/risk-neutral probabilities commonly employed in derivative asset pricing theory. We supply an example in which our characterization holds while the standard representation of state-price densities in terms of marginal utilities of optimal consumption fails.