 
Summary: The Annals of Probability
2006, Vol. 34, No. 5, 18641909
DOI: 10.1214/009117906000000359
© Institute of Mathematical Statistics, 2006
SINGULAR CONTROL WITH STATE CONSTRAINTS ON
UNBOUNDED DOMAIN
BY RAMI ATAR1 AND AMARJIT BUDHIRAJA2
TechnionIsrael Institute of Technology and University of North Carolina
We study a class of stochastic control problems where a cost of the form
E
[0,)
es[ (Xs)ds + h(Y
s )dYs](0.1)
is to be minimized over control processes Y whose increments take values in
a cone Y of Rp, keeping the state process X = x +B +GY in a cone X of Rk,
k p. Here, x X, B is a Brownian motion with drift b and covariance ,
G is a fixed matrix, and Y is the RadonNikodym derivative dY/dY. Let
L = (1/2)trace( D2)  b · D where D denotes the gradient. Solutions to
the corresponding dynamic programming PDE,
[(L + )f  ] sup
