Envelope theorems in Banach lattices

We derive envelope theorems for optimization problems in which the value function takes values in a general Banach lattice, and not necessarily in the real line. We impose no restriction whatsoever on the choice set. Our result extend therefore the ones of Milgrom and Segal (2002). We apply our results to discuss the existence of a well-defined notion of marginal utility of wealth in optimal consumption-portfolio problems in which the utility from consumption is additive but possibly state-dependent and, most importantly, the information structure is not required to be Markovian. In this general setting, the value function is itself a random variable and, if integrable, takes values in a Banach lattice so that our general results can be applied.

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