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A martingale approach to optimal portfolios with jump-diffusions. (English) Zbl 1251.91055

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 [I. Karatzas and 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.

MSC:

91G10 Portfolio theory
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
91B16 Utility theory
60J75 Jump processes (MSC2010)
60J60 Diffusion processes

Citations:

Zbl 0941.91032
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