Facelifting in utility maximization
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We establish the existence and characterization of a primal and a dual facelift—discontinuity of the value function at the terminal time—for utility maximization in incomplete semimartingale-driven financial markets. Unlike in the lower and upper hedging problems, and somewhat unexpectedly, a facelift turns out to exist in utility maximization despite strict convexity in the objective function. In addition to discussing our results in their natural, Markovian environment, we also use them to show that the dual optimizer cannot be found in the set of countably additive (martingale) measures in a wide variety of situations.
KeywordsBoundary layer Convex analysis Convex duality Facelift Financial mathematics Incomplete markets Markov processes Utility maximization Unspanned endowment
Mathematics Subject Classification (2010)91G10 91G80 60K35
JEL ClassificationC61 G11
The authors would like to thank Mihai Sîrbu, Kim Weston, and the two anonymous referees for their many constructive comments.