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.
@article{bwmeta1.element.bwnjournal-article-div17i1-2n9bwm,
author = {Benoit Truong-Van and Truong Xuan Duc Ha},
title = {Existence of viable solutions for a nonconvex stochastic differential inclusion},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {17},
year = {1997},
pages = {107-131},
zbl = {0910.60050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-div17i1-2n9bwm}
}
Benoit Truong-Van; Truong Xuan Duc Ha. Existence of viable solutions for a nonconvex stochastic differential inclusion. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 17 (1997) pp. 107-131. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-div17i1-2n9bwm/
[000] [1] J.P. Aubin, G. Da Prato, Stochastic viability and invariance, Annali Scuola Normale di Pisa, 27 (1990), 595-614. | Zbl 0741.60046
[001] [2] J.P. Aubin, G. Da Prato, Stochastic Nagumo's viability theorem, Stochastic Analysis and Applications, 13 (1995), 1-11. | Zbl 0816.60053
[002] [3] N. Dunford, J.T. Schwartz, Linear Operators, Part I, Interscience Publisher Inc., New York 1957. | Zbl 0128.34803
[003] [4] S. Gautier, L.Thibault, Viability for constrained stochastic differential equations, Differential and Integral Equations 6 (6) (1993), 1395-1414. | Zbl 0780.93085
[004] [5] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, New York 1988. | Zbl 0638.60065
[005] [6] M. Kisielewicz, Viability theorem for stochastic inclusions, Discussiones Mathematicae - Differential Inclusions 15 (1995), 61-74.
[006] [7] A. Milian, A note on the stochastic invariance for Itô equations, Bulletin of the Polish Academy of Sciences Mathematics, 41 (1993), 139-150. | Zbl 0796.60071
[007] [8] X.D.H. Truong, Existence of viable solutions of nonconvex-valued differential inclusions in Banach spaces, Portugalae Mathematica, 52 (1995), 241-250. | Zbl 0824.34021
[008] [9] X.D.H. Truong, An existence result for nonconvex viability problem in Banach spaces, Preprint N.16 (1996), University of Pau, France.
[009] [10] Qi Ji Zhu, On the solution set of differential inclusions in Banach spaces, J. Differential Equations, 93 (2) (1991), 213-236.