Extended weak convergence and utility maximisation with proportional transaction costs
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Extended weak convergence and utility maximisation with proportional transaction costs Erhan Bayraktar1 · Leonid Dolinskyi2 · Yan Dolinsky3
Received: 18 December 2019 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study utility maximisation with proportional transaction costs. Assuming extended weak convergence of the underlying processes, we prove the convergence of the time-0 values of the corresponding utility maximisation problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended weak convergence theory developed in Aldous (Weak Convergence of Stochastic Processes for Processes Viewed in the Strasbourg Manner, 1981) and on the Meyer–Zheng topology introduced in Meyer and Zheng (Ann. Inst. Henri Poincaré Probab. Stat. 20:353–372, 1984). Keywords Utility maximisation · Proportional transaction costs · Extended weak convergence · Meyer–Zheng topology Mathematics Subject Classification (2010) 91B16 · 91G10 · 60F05 JEL Classification G11 · C65 E. Bayraktar is supported in part by the National Science Foundation under grant DMS-1613170 and in part by the Susan M. Smith Professorship. Y. Dolinsky is supported in part by the GIF Grant 1489-304.6/2019 and the ISF grant 160/17.
B Y. Dolinsky
[email protected] E. Bayraktar [email protected] L. Dolinskyi [email protected]
1
Department of Mathematics, University of Michigan, Ann Arbour, USA
2
Department of Economic Cybernetics, The University of the State Fiscal Service, Kiev, Ukraine
3
Department of Statistics, Hebrew University, Jerusalem, Israel
E. Bayraktar et al.
1 Introduction We deal with the continuity of the utility maximisation problem in the presence of proportional transaction costs under convergence in distribution of the financial markets. More specifically, we focus on utility maximisation from terminal wealth under an admissibility condition on the wealth processes. In the presence of transaction costs, the problem of utility maximisation goes back to Magill and Constantinides [22], where the authors considered maximising the expected consumption in the Black–Scholes model. Analysing this as a stochastic control problem, Davis and Norman [12] gave a rigorous proof for the heuristic derivation of Magill and Constantinides. Later, Shreve and Soner [27] removed technical conditions needed in [12] and provided a complete solution under the assumption that the value function is finite. The more tractable problems of maximising the asymptotic growth rate for logarithmic or power utility in the Black–Scholes model under transaction costs have been studied by Taksar et al. [29] and Dumas and Luciano [16]. In [21], Kallsen and Muhle-Karbe applied the shadow price approach and solved the Merton problem with logarithmic utility and proportional transaction costs. Later this approach was extended to the power utility case; see Gerhold et al. [17]. Cvitani´c and Karatzas [10] were the first to establish continuous-tim
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