Utility maximization on the real line under proportional transaction costs
type: Item
instance of: scholarly article
title: Utility maximization on the real line under proportional transaction costs
zbMATH Open document ID: 1039.91019
author: Bruno Bouchard
published in: Finance and Stochastics
publication date: 2004-03-16T00:00:00Z
review text: The author considers a multivariate financial market with transaction costs as in \textit{Yu. M. Kabanov} and \textit{G. Last} [Math. Finance 12, No. 1, 63--70 (2002; Zbl 1008.91049)]. Let \(x\in\mathbb R^d\) be an initial endowment, let \(L\) be a trading strategy, and let \(X_T^{x,L}\) be the value at time \(T\) of the induced portfolio process. Given a random claim \(G\), the liquidation value of the terminal portfolio after delivering \(G\) is given by \(\ell(X_T^{x,L}-G)\), where \(\ell\) stands for the liquidation function, i.e. \(\ell(x)\) is the maximal cash endowment that we can get from portfolio \(x\) when creating all the positions in risky assets and paying the transaction costs. In this paper the author considers the problem of maximizing the expected utility of \(\ell(X_T^{x,L}-G)\), i.e. \[ V(x):=\sup_{L} EU \left(\ell(X_T^{x,L}-G)\right) \] for a utility function \(U:\mathbb R \to\mathbb R\). For the univariate financial market (only one risky assert) with \(G=0\), \textit{J. Cvitanić} and \textit{H. Wang} [J. Math. Econ. 35, No. 2, 223--231 (2001; Zbl 1056.91028)] proved that existence holds in the maximization problem for utility functions defined on \((0,\infty)\) by using the asymptotic elasticity condition introduced by \textit{D. Kramkov} and \textit{W. Schachermayer} [Ann. Appl. Probab. 9, No. 3, 904--950 (1999; Zbl 0967.91017)], \[ AE_+(U):=\limsup_{r\uparrow\infty} \frac{rU'(r)}{U(r)}<1. \] For incomplete financial markets \textit{W. Schachermayer} [Ann. Appl. Probab. 11, No. 3, 694--734 (2001; Zbl 1049.91085)] proved that a solution exists under the additional condition \[ AE_-(U):=\liminf_{r\downarrow \infty}\frac {rU'(r)} {U(r)}>1. \] In this article this result is extended to the multivariate financial markets with proportional transaction costs and \(G\in L^{\infty}\) and with a utility function defined on \(\mathbb R\) and satisfying the reasonable asymptotic elasticity conditions: \[ AE_+(U):=\limsup_{r\uparrow\infty}\frac{rU'(r)}{U(…
reviewed by: M. P. Moklyachuk
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 91B16
Mathematics Subject Classification ID: 91B24
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 2059095
zbMATH Keywords: transaction costs
zbMATH Keywords: utility maximization
zbMATH Keywords: reasonable asymptotic elasticity
zbMATH Keywords: hedging
zbMATH Keywords: option pricing
MaRDI profile type: Publication
full work available at URL: s007800200068
OpenAlex ID: W2080307616
DOI: 10.1007/S007800200068
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description: scientific article; zbMATH DE number 2059095