Appl Math Optim 53:279–309 (2006)
2006 Springer Science+Business Media, Inc.
Optimal Investment in a L´evy Market
Jos´e Manuel Corcuera,
and Wim Schoutens
Facultat de Matem`atiques, Universitat de Barcelona,
Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain
CEMAPRE and ISEG,
Rua do Quelhas 6, 1200-781 Lisboa, Portugal
Katholieke Universiteit Leuven,
U.C.S., W. De Croylaan 54, B-3001 Leuven, Belgium
Abstract. In this paper we consider the optimal investment problem in a market
where the stock price process is modeled by a geometric L´evy process (taking
into account jumps). Except for the geometric Brownian model and the geometric
Poissonian model, the resulting models are incomplete and there are many equivalent
martingale measures. However, the model can be completed by the so-called power-
jump assets. By doing this we allow investment in these new assets and we can
try to maximize the expected utility of these portfolios. As particular cases we
obtain the optimal portfolios based in stocks and bonds, showing that the new assets
are superﬂuous for certain martingale measures that depend on the utility function
Key Words. Portfolio optimization, L´evy processes, Martingale method, Repli-
cating portfolios, Incomplete markets, HARA utility.
AMS Classiﬁcation. 60H30, 60G46, 91B28.
W. Schoutens is a Postdoctoral Fellow of the Fund for Scientiﬁc Research–Flanders (Belgium) (F.W.O.
- Vlaanderen). The work of Jos´e Manuel Corcuera and David Nualart is supported by the MCyT Grant No.
BFM2003-04294. The work of Jo˜ao Guerra is also supported by this grant and by FCT-POCTI.