Utility maximization on the real line under proportional transaction costs
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: \[ AE_+(U):=\limsup_{r\uparrow\infty}\frac{rU'(r)}{U(r)}<1,\quad AE_-(U):=\liminf_{r\downarrow\-\infty}\frac{rU'(r)}{U(r)}>1. \] It is proved that existence and duality hold! in the class of targets that can be approximated by bounded from below strategies. Under some additional conditions it is proved that the optimal target is indeed attainable. As an application a dual formulation for the exponential reservation price is proposed.
scientific article; zbMATH DE number 2059095