Appl Math Optim 49:27–41 (2004)
2003 Springer-Verlag New York Inc.
A Connection between Singular Stochastic Control
and Optimal Stopping
Fred Espen Benth
and Kristin Reikvam
Centre of Mathematics for Applications,
Department of Mathematics, University of Oslo,
P.O. Box 1053, Blindern, N-0316 Oslo, Norway
Department of Economics and Business Administration,
Agder University College,
Serviceboks 422, N-4604 Kristiansand, Norway
Abstract. We show that the value function of a singular stochastic control problem
is equal to the integral of the value function of an associated optimal stopping
problem. The connection is proved for a general class of diffusions using the method
of viscosity solutions.
Key Words. Singular stochastic control, Optimal stopping, Variational inequali-
ties, Viscosity solutions.
AMS Classiﬁcation. 93E20, 60G40, 49L25, 60G35.
If V (t, x) is the value function of a stochastic singular control problem, Bather and Cher-
noff  observed that ∂ V (t, x)/∂ x is the value function of an associated optimal stopping
problem. They studied a monotone follower problem for Brownian motion, which was
later treated by Benes et al.  and Karatzas , . Using probabilistic methods, the
connection between singular control and optimal stopping was established rigorously
by Karatzas and Shreve ,  for monotone and reﬂected follower problems where
FEB acknowledges ﬁnancial support from MaPhySto - Center for Mathematical Physics and Stochas-
tics, University of Aarhus. MaPhySto is funded by a grant from the Danish National Research Foundation.
KR acknowledges ﬁnancial support by the Norwegian Research Council under Grant NFR 118868/410.