Appl Math Optim 55:93–122 (2007)
2006 Springer Science+Business Media, Inc.
Optimal Long-Term Investment Model with Memory
and Yumiharu Nakano
Department of Mathematics, Faculty of Science, Hokkaido University,
Sapporo 060-0810, Japan
Center for the Study of Finance and Insurance, Osaka University,
Toyonaka 560-8531, Japan
Communicated by B. Øksendal
Abstract. We consider a ﬁnancial market model driven by an R
process with stationary increments which is different from Brownian motion. This
driving-noise process consists of n independent components, and each component
has memory described by two parameters. For this market model, we explicitly solve
optimal investment problems. These include: (i) Merton’s portfolio optimization
problem; (ii) the maximization of growth rate of expected utility of wealth over the
inﬁnite horizon; (iii) the maximization of the large deviation probability that the
wealth grows at a higher rate than a given benchmark. The estimation of parameters
is also considered.
Key Words. Optimal investment, Long-term investment, Processes with memory,
Processes with stationary increments, Riccati equations, Large deviations.
AMS Classiﬁcation. Primary 91B28, 60G10, Secondary 62P05, 93E20.
In this paper we study optimal investment problems for a ﬁnancial market model with
memory. This market model M consists of n risky and one riskless assets. The price
The work of the second author was partially supported by a Research Fellowship of the Japan Society
for the Promotion of Science for Young Scientists.