Risk sensitive asset management with transaction costs
The mathematical problem of optimally managing a portfolio of securities when there are transaction costs has received considerable research attention in recent years. For the classical problem, where the objective is to maximize expected utility of terminal wealth, most of the attention has been devoted to the case of proportional costs, that is, to the case where the cost associated with a transaction is proportional to the amount of money that is shifted between the securities. Typically, the optimal strategy is characterized by a no-trade region with trading that is essentially continuous on its boundary used to keep a certain process contained in the region. Other approaches assumed the transaction cost has a fixed component, thereby precluding the optimality of continuous trading. Some authors applied the theories of optimal stopping and quasi-variational inequalities to two portfolio management problems, both involving two assets (a bank account with interest rate zero and a geometric Brownian motion stock) and transaction costs. In all of this researches the assets are modelled with conventional stochastic differential equations. The only physical processes being modeled are the assets themselves along with the bank account and its short term interest rate. Moreover, there is no explicit dependence of the asset processes on this interest rate. Meanwhile, several authors have developed transaction-free optimal portfolio models where the asset processes explicitly depend on underlying economic factors which are also explicitly modeled with stochastic differential equations. This paper develops a continuous time risk-sensitive portfolio optimization model with a general transaction cost structure and where the individual securities or asset categories are explicitly affected by underlying economic factors. The security prices and factors follow diffusion processes with the drift and diffusion coefficients for the securities being functions of the factor levels. The main result in this paper is the characterization of optimal trading strategies in terms of the security prices and the factors levels and of what authors call ``Risk Sensitive Quasi-Variational Inequalities'' (RS-QVI). That is, it can be computed by solving an RS-QVI. The Kelly criterion case is also studied and the various results are related to the recent work by \textit{A. J. Morton} and \textit{R. S. Pliska} [Math. Finance 5, No. 4, 337-356 (1995; Zbl 0866.90020)].
risk-sensitive impulsive stochastic control
scientific article; zbMATH DE number 1452412