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Utility Maximization Problem with Transaction Costs: Optimal Dual Processes and Stability

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Abstract

In this paper, we investigate the stability problem of the numéraire-based utility maximization problem in markets with transaction costs, where the stock price is not necessarily a semimartingale. Precisely, the static stability of primal and dual value functions as well as the convergence of primal and dual optimizers are presented when perturbations occur in the utility function and in the physical probability. Furthermore, this study focuses on the optimal dual process (ODP), which induces the dual optimizer and attains optimality for a dynamical dual problem. Properties of ODPs are discussed which are complement of the duality theory for this utility maximization problem. When the parameters of the market and the investor are slightly perturbed, both the dual optimizer and the associated optimal dual process are stable. Thus, a shadow price process is constructed based on the sequence of ODPs corresponding to problems with small misspecified parameters.

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Notes

  1. These assumptions avoid special notations for possible trading at the terminal time T. In this case, we may assume without loss of generality that the agent liquidates her position in the stock shares at time T. For more details see e.g.  [10, Remark 4.2] or [14, p. 1895].

  2. A real-valued optional process \(X =(X_t)_{0 \le t \le T}\) is called optional strong supermartingale if

    • \(X_{\tau }\) is integrable for every [0, T]-valued stopping time \(\tau \);

    • For all stopping times \(\sigma \) and \(\tau \) with \( 0 \le \sigma < \tau \le T\), we have \( X_{\sigma } \ge {\mathbb {E}}\left[ \left. {X_{\tau }}\right| {{\mathcal {F}}_{\sigma }}\right] . \)

    It is a generalisation of càdlàg supermartingales. See [38] and [21, Appendix I] for more properties.

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Acknowledgements

The authors gratefully acknowledge financial support from National Natural Science Foundation of China, Youth Science Fund Project under Grant 11801365 and 11901097, from Educational research projects for young and middle-aged teachers in Fujian under grant JT180073, from the research fund of Fujian Engineering Research Center of Public Service Big Data Mining and Application, from the Austrian Science Fund (FWF) under Grant P25815, from the European Research Council under European Research Council (ERC) Advanced Grant 321111 and from the University of Vienna under short-term grand abroad (KWA). This work was partially completed during the visit of L. Gu and J. Yang at the Centre de Mathématiques Appliqées, École Polytechnique, hosted by Prof. N. Touzi, who are gratefully acknowledged. The comments from the anonymous referees are greatly appreciated.

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Authors and Affiliations

  1. College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, China

    Lingqi Gu

  2. School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yiqing Lin

  3. Fakultät für Mathematik und Geoinformation, TU Wien, 1040, Vienna, Austria

    Junjian Yang

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  1. Lingqi Gu
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  2. Yiqing Lin
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Correspondence to Yiqing Lin.

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Gu, L., Lin, Y. & Yang, J. Utility Maximization Problem with Transaction Costs: Optimal Dual Processes and Stability. Appl Math Optim 84, 1903–1922 (2021). https://doi.org/10.1007/s00245-020-09699-8

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