Abstract
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.
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Acknowledgements
The authors would like to thank Mihai Sîrbu, Kim Weston, and the two anonymous referees for their many constructive comments.
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During the preparation of this work, the first author has been supported by the National Science Foundation under Grant No. DMS-1411809 (2014–2017). The second author has been supported by the Swiss National Foundation through grant SNF \(200021\_153555\) and by the Swiss Finance Institute. The third author has been supported by the National Science Foundation under Grants No. DMS-0706947 (2010–2015) and No. DMS-1107465 (2012–2017). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).
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Larsen, K., Soner, H.M. & Žitković, G. Facelifting in utility maximization. Finance Stoch 20, 99–121 (2016). https://doi.org/10.1007/s00780-015-0274-y
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DOI: https://doi.org/10.1007/s00780-015-0274-y
Keywords
- Boundary layer
- Convex analysis
- Convex duality
- Facelift
- Financial mathematics
- Incomplete markets
- Markov processes
- Utility maximization
- Unspanned endowment