Discussiones Mathematicae Differential Inclusions 17 (1997) 107-131



Benoit Truong-Van

Laboratoire de Mathématiques Appliquées
URA-CNRS 1204 Université de Pau, France

Truong Xuan Duc Ha

Hanoi Institute of Mathematics
P.O. Box 631, Boho, Hanoi, Vietnam


For the stochastic viability problem of the form

dx(t) ∈ F(t,x(t))dt+g(t,x(t))dW(t), x(t) ∈ K(t),

where K, F are set-valued maps which may have nonconvex values, g is a single-valued function, we establish the existence of solutions under the assumption that F and g possess Lipschitz property and satisfy some tangential conditions.

Keywords: Stochastic differential inclusion, viable solution, tangential condition, Lipschitz property.

1991 Mathematics Subject Classifications: 60H10, 34A60.


[1] J.P. Aubin, G. Da Prato, Stochastic viability and invariance, Annali Scuola Normale di Pisa, 27 (1990), 595-614.
[2] J.P. Aubin, G. Da Prato, Stochastic Nagumo's viability theorem, Stochastic Analysis and Applications, 13 (1995), 1-11.
[3] N. Dunford, J.T. Schwartz, Linear Operators, Part I, Interscience Publisher Inc., New York 1957.
[4] S. Gautier, L.Thibault, Viability for constrained stochastic differential equations, Differential and Integral Equations 6 (6) (1993), 1395-1414.
[5] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, New York 1988.
[6] M. Kisielewicz, Viability theorem for stochastic inclusions, Discussiones Mathematicae - Differential Inclusions 15 (1995), 61-74.
[7] A. Milian, A note on the stochastic invariance for Itô equations, Bulletin of the Polish Academy of Sciences Mathematics, 41 (1993), 139-150.
[8] X.D.H. Truong, Existence of viable solutions of nonconvex-valued differential inclusions in Banach spaces, Portugalae Mathematica, 52 (1995), 241-250.
[9] X.D.H. Truong, An existence result for nonconvex viability problem in Banach spaces, Preprint N.16 (1996), University of Pau, France.
[10] Qi Ji Zhu, On the solution set of differential inclusions in Banach spaces, J. Differential Equations, 93 (2) (1991), 213-236.

Received 15 December 1997
Revised 19 March 1998