Appl Math Optim 56:67–103 (2007)
2007 Springer Science+Business Media, Inc.
Impulsive Control of Portfolios
Jan Palczewski and
Institute of Mathematics, Polish Academy of Sciences,
Sniadeckich 8, 00-950 Warszawa, Poland
Abstract. In this paper a general model of a market with asset prices and econom-
ical factors of Markovian structure is considered. The problem is to ﬁnd optimal
portfolio strategies maximizing a discounted inﬁnite horizon reward functional con-
sisting of an integral term measuring the quality of the portfolio at each moment and
a discrete term measuring the reward from consumption. There are general trans-
action costs which, in particular, cover ﬁxed plus proportional costs. It is shown,
under general conditions, that there exists an optimal impulse strategy and the value
function is a solution to the Bellman equation which corresponds to suitable quasi-
Key Words. Markov process, Impulsive control, Bellman equation, Portfolio
optimization, Transaction costs.
AMS Classiﬁcation. 93E20, 91B28.
On a given probability space (, F, P) with ﬁltration (F
consider a market modeled
by a time homogeneous Markov process (S(t), X (t))
, where S(t) = (S
(t)) ∈ (0, ∞)
denotes the prices of d assets and X (t) ∈ R
stands for m economic
factors. The use of economic factors was fully justiﬁed in . Firstly, it expands the
capabilities of the model to some extent. Secondly, it facilitates and improves the quality
of statistical estimation of model parameters. We assume that (S(t), X (t)) has right
continuous and left limited trajectories and satisﬁes the so-called Feller property, i.e. its
The ﬁrst author’s research was supported by Grant KBN 1-P03A-012-28.