Abstract
In this paper, we study expected utility maximization under ratchet and drawdown constraints on consumption in a general incomplete semimartingale market using duality methods. The optimization is considered with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains, we introduce a natural extension of the notion of running maximum to arbitrary nonnegative optional processes and study its properties. The dual domains for optimization are characterized in terms of solidity with respect to an ordering that is introduced on the set of nonnegative optional processes. The abstract duality result we obtain for the optimization problem is used in order to derive a more detailed characterization of solutions in the complete market case.
Funding Statement
The research was supported in part by the National Science Foundation under Grant DMS-1908903. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Part of this research was performed while the author was visiting the Institute for Mathematical and Statistical Innovation (IMSI), which is supported by the National Science Foundation (Grant DMS-1929348).
Acknowledgments
I would like to thank my advisor Mihai Sîrbu for the many helpful discussions on topics related to this paper. The comments of the anonymous referees led to major improvements of the paper and are greatly appreciated. I would also like to thank Bahman Angoshtari for several insightful discussions on ratchet and drawdown constraints.
Citation
Anastasiya Tanana. "Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets." Ann. Appl. Probab. 33 (5) 4127 - 4162, October 2023. https://doi.org/10.1214/22-AAP1918
Information
