Optimal portfolios in Lévy markets under state-dependent bounded utility functions
The authors consider Merton's portfolio optimization problem in a specific but popular non-Markovian market model driven by a Lévy process. They follow the usual dual variational approach to show that the domain of the dual problem enjoys an explicit parametrization built on multiplicative optional decomposition. In their case, the boundedness and potential differentiability of the utility function cause some technical subtleties. For example, the dual optimal process can be 0 with positive probability, thus the representation of Kunita does not apply anymore. A key element that the authors use to overcome these technical difficulties is an exponential representation theorem for negative semimartingales by Föllmer and Kramkov. This result leads to an explicit characterization of the dual domain, based on those nonnegative supermartingales that can be written as stochastic exponentials. To validate this approach, they prove a closure property for integrals with respect to a fixed compensated Poisson random measure, a result of interest on its own. Finally, unlike some previous works on the subject, the authors do not use the so-called bipolar theorem of Kramkov and Schachermayer to argue the attainability of the optimal final wealth. Instead, they rely on the fundamental characterization of contingent claims that are replicable, reducing the problem of finding the optimal primal solution to a super-replication problem. The dual problem proposed in this paper offers some advantages. Since the dual class enjoys a fairly explicit description and parametrization, their results could be considered as the first step towards a feasible computational implementation of the convex duality method. Furthermore, the specific results obtained for the Lévy market can be used to characterize the elements of the dual domain and the admissible trading strategies. In particular, in some specific cases they show that the dual solution is a risk-neutral martingale.
scientific article; zbMATH DE number 5701576
scientific article