A new class of strategies and application to utility maximization for unbounded processes

The problem of maximization of expected utility from terminal wealth relies on the definition of admissible strategies, the ones which the agents are allowed to trade in. It is known that the classical definition of admissibility can lead to a trivial maximization if the price process is not locally bounded. Hence we introduce a suitable enlargement of the classical set and we show its properties, highlighting economic significance. We formulate and solve the utility maximization with this new class of strategies, providing also some concrete examples. Acknowledgments: The author would like to express her gratitude to Prof. M. Frittelli and to Prof. S. Herzel for advice and support. 1 General admissible strategies 1.1 Setting and recalls Our setting is the general semimartingale model of a financial market as defined by Delbaen and Schachermayer in [4]. Let (Ω,F , (Ft)t∈[0,T ], P ) be a filtered probability space, where we assume that the filtration satisfies the usual assumptions of right continuity and completeness. T is the horizon, which can be as well +∞: if this is the case, when considering processes Y it is understood that Y∞ = limt Yt exists. The Rd−valued càd-làg semi-martingale X = (Xt)t∈[0,T ] represents the (discounted) price process of d tradeable assets. An Rd−valued predictable process H = (Ht)t∈[0,T ] is called a trading strategy simply if H is X−integrable. It common knowledge that some ∗Dipartimento di Economia Sezione di Finanza Matematica. Università degli Studi di Perugia, via A. Pascoli 20. 06123 Perugia, Italy. Fax: +39-075 5855221. Email: [email protected]