Appl Math Optim 55:359–384 (2007)
2007 Springer Science+Business Media, Inc.
Stopping Problems of Certain Multiplicative Functionals
and Optimal Investment with Transaction Costs
Department of Mathematical Science, Graduate School of Engineering Science,
Osaka University, Toyonaka 560-8531, Japan
Communicated by A. Bensoussan
Abstract. Optimal stopping and impulse control problems with certain multiplica-
tive functionals are considered. The stopping problems are solved by showing the
unique existence of the solutions of relevant variational inequalities. However, since
functions deﬁning the multiplicative costs change the signs, some difﬁculties arise
in solving the variational inequalities. Through gauge transformation we rewrite the
variational inequalities in different forms with the obstacles which grow exponen-
tially fast but with positive killing rates. Through the analysis of such variational
inequalities we construct optimal stopping times for the problems. Then optimal
strategies for impulse control problems on the inﬁnite time horizon with multi-
plicative cost functionals are constructed from the solutions of the risk-sensitive
variational inequalities of “ergodic type” as well. Application to optimal investment
with ﬁxed ratio transaction costs is also considered.
Key Words. Optimal stopping, Impulse control, Transaction costs, Variational
inequalities, Multiplicative functionals.
AMS Classiﬁcation. 91B28, 91B70, 49N25, 62L15, 35J85, 60G40.
In the study of optimal stopping problems under Markovian settings considered so far,
criterion functions are deﬁned mostly by additive functionals (see  and the references
This research was supported in part by Grant-in-Aid for Scientiﬁc Research No. 16340025, JSPS.