A martingale approach to optimal portfolios with jump-diffusions
The paper studies optimal trading strategies for the problems of maximizing expected utility of consumption and terminal wealth under multi-dimensional jump-diffusion models. Evidently, the market is incomplete. However, the approach of the combination of martingale and duality techniques, originated by [\textit{I. Karatzas} and \textit{S. E. Shreve}, Methods of mathematical finance. Applications of Mathematics. Berlin: Springer. (1998; Zbl 0941.91032)] in order to study the similar problem under a pure diffusion process in a complete market, is modified and applied. It is proved that the optimal consumption process and the optimal trading strategy are determined by the martingale measure whose parameter is a solution to a system of nonlinear equations. This is in a contrast to the standard duality approach. The modification, proposed in the paper, has the advantage that the optimal martingale measure as well as the optimal consumption process and trading strategy can be directly obtained by solving the system of nonlinear equations. As an example, HARA utilities are studied when the parameters in the model are deterministic functions.
scientific article; zbMATH DE number 6039299
scientific article