Appl Math Optim 40:355–375 (1999)
1999 Springer-Verlag New York Inc.
Stochastic Control Problems where Small Intervention
Costs Have Big Effects
Department of Mathematics, University of Oslo,
P. O. Box 1053, Blindern, 0316 Oslo, Norway
Institute of Finance and Management Science,
Norwegian School of Economics and Business Administration,
Helleveien 30, N-5035 Bergen-Sandviken, Norway
Abstract. We study an impulse control problem where the cost of interfering in a
stochastic system with an impulse of size ζ ∈ R is given by
c + λ|ζ|,
where c and λ are positive constants. We call λ the proportional cost coefﬁcient and
c the intervention cost. We ﬁnd the value/cost function V
for this problem for each
c > 0 and we show that lim
= W, where W is the value function for the
corresponding singular stochastic control problem. Our main result is that
=∞ at c = 0.
This illustrates that the introduction of an intervention cost c > 0, however small,
into a system can have a big effect on the value function: the increase in the value
function is in no proportion to the increase in c (from c = 0).
Key Words. Impulse control, Vanishing intervention cost, Quasi-variational in-
equalities, Singular stochastic control, Nonrobustness feature.
AMS Classiﬁcation. 93E20, 60G40, 60J65, 49J40, 35R35.