Appl Math Optim 50:21–65 (2004)
2004 Springer-Verlag New York, LLC
Optimal Control Problem Associated with Jump Processes
Department of Mathematics,
Faculty of Science, Ehime University,
Matsuyama Ehime 7908577, Japan
Communicated by B. Øksendal
Abstract. An optimal portfolio/control problem is considered for a two-dimen-
sional model in ﬁnance. A pair consisting of the wealth process and cumulutative
consumption process driven by a geometric L´evy process is controlled by adapted
processes. The value function appears and turns out to be a viscosity solution to
some integro-differential equation, by using the Bellman principle.
Key Words. Stochastic control of jump type, Mathematical ﬁnance, Viscosity
solution of PDE.
AMS Classiﬁcation. Primary 60H30, Secondary 49J22, 90A09.
There are several articles which suggest that some so-called heavy tail distributions ﬁt
real data in ﬁnance well, and the importance of jump-type price processes which are
For example, Eberlein  suggests the importance of generalized hyperbolic (GH)
distribution in empirical date. Barndorff-Nielsen  has pointed out the importance of
heavy tail distributions such as (generalized) normal inverse Gaussian (GIG) distribu-
tions, in the form of exponential stochastic models driven by L´evy processes whose
marginal distributions are heavy tail. Several models in mathematical ﬁnance based on
this distribution or on GH distributions have been proposed with a considerable ﬁt. See
 for example.