Appl Math Optim 54:71–93 (2006)
2006 Springer Science+Business Media, Inc.
Impulse Control of One-Dimensional Itˆo Diffusions with an
Expected and a Pathwise Ergodic Criterion
Andrew Jack and Mihail Zervos
Department of Mathematics, King’s College London,
The Strand, London WC2R 2LS, England
Abstract. We consider the problem of controlling a general one-dimensional Itˆo
diffusion by means of an impulse control process. The objective is to minimise a
long-term expected criterion as well as a long-term pathwise criterion that penalise
both deviations of the state process from a given nominal point and the use of
impulsive control effort. In particular, each time the controller deploys an impulse
to reposition the system’s state, a ﬁxed cost and a cost proportional to the impulse’s
size are incurred. We solve the resulting optimisation problems and we provide an
explicit characterisation of an optimal control strategy under general assumptions.
The control of a foreign exchange rate or an inﬂation rate presents a potential
application of the model that we study.
Key Words. Itˆo diffusions, Impulse control, Ergodic criterion.
AMS Classiﬁcation. 93E20, 49N25, 49J40.
We consider a stochastic system, the state of which is modelled by the controlled, one-
dimensional Itˆo diffusion
) dt + dZ
= x ∈ R,
where W is a standard, one-dimensional Brownian motion. The controlled process Z
is piecewise constant: the jumps of this process occur at the times when the system’s
This research was supported by EPSRC Grant No. GR/S22998/01.