Appl Math Optim 49:99–112 (2004)
2004 Springer-Verlag New York Inc.
Stochastic Control with Exit Time and Constraints,
Application to Small Time Attainability of Sets
and Aurel R˘a¸scanu
Laboratoire de Math´ematiques,
Unit´e CNRS FRE 2218, Universit´e de Bretagne Occidentale,
6 Avenue Victor Le Gorgeu, B.P. 809, 29285-Brest cedex, France
Facultatea de matematic˘a, Universitatea “Alexandru Ioan Cuza”,
6600 - Ia¸si, Romania
Communicated by A. Bensoussan
Abstract. We study the existence of a solution of controlled stochastic differential
equations remaining in a given set of constraints at any time smaller than the exit
time of a given open set. We also investigate the small time attainability of a given
closed set K , i.e., the property that, for all arbitrary small time horizon T and for all
initial condition in a sufﬁciently small neighborhood of K , there exists a solution to
the controlled stochastic differential equation which reaches K before T .
Key Words. Stochastic viability, Controllability.
AMS Classiﬁcation. 93E03, 60H10, 60H30, 35K65, 35D05, 49L25.
Given a d-dimensional Brownian motion W on a probability space (, F, P), we con-
sider the following stochastic control problem:
(t) = b(X
(t), u(t)) dt + σ(X
(t), u(t)) dW(t), t ∈ [0,+∞),
(0) = x ∈ R