Appl Math Optim 38:327–352 (1998)
1998 Springer-Verlag New York Inc.
A Pair of Explicitly Solvable Singular Stochastic
M. H. A. Davis
and M. Zervos
Department of Electrical and Electronic Engineering,
Imperial College of Science, Technology and Medicine,
London SW7 2BT, England
Department of Statistics, School of Mathematics and Statistics,
University of Newcastle, Newcastle upon Tyne NE1 7RU, England
Abstract. We consider a general model of singular stochastic control with inﬁnite
time horizon and we prove a “veriﬁcation theorem” under the assumption that the
case, under the assumption that the HJB equation has a solution in W
p ≥ 1, we prove a very general “veriﬁcationtheorem” by employing the generalized
Meyer–Itˆochange of variablesformulawithlocaltimes. In whatfollows,weconsider
two special cases which we explicitly solve. These are the formal equivalent of the
one-dimensional inﬁnite time horizon LQG problem and a simple example with
radial symmetry in an arbitrary Euclidean space. The value function of either of
these problems is C
and is expressed in terms of specialfunctions, and, in particular,
the conﬂuent hypergeometric function and the modiﬁed Bessel function of the ﬁrst
Key Words. Singular stochastic control, Variational inequality, Local times, Con-
ﬂuent hypergeometric function, Modiﬁed Bessel functions.
AMS Classiﬁcation. Primary 93E20, Secondary 33C10, 33C15, 33C90.
This work was supported by the UK Defence Research Agency under Agreement No. 2037/393/RAE.