# Existence of viable solutions for a nonconvex stochastic differential inclusion

Benoit Truong-Van; Truong Xuan Duc Ha

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1997)

- Volume: 17, Issue: 1-2, page 107-131
- ISSN: 1509-9407

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topBenoit Truong-Van, and Truong Xuan Duc Ha. "Existence of viable solutions for a nonconvex stochastic differential inclusion." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 17.1-2 (1997): 107-131. <http://eudml.org/doc/275927>.

@article{BenoitTruong1997,

abstract = {
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.
},

author = {Benoit Truong-Van, Truong Xuan Duc Ha},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {Stochastic differential inclusion; viable solution; tangential condition; Lipschitz property; stochastic differential inclusion},

language = {eng},

number = {1-2},

pages = {107-131},

title = {Existence of viable solutions for a nonconvex stochastic differential inclusion},

url = {http://eudml.org/doc/275927},

volume = {17},

year = {1997},

}

TY - JOUR

AU - Benoit Truong-Van

AU - Truong Xuan Duc Ha

TI - Existence of viable solutions for a nonconvex stochastic differential inclusion

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 1997

VL - 17

IS - 1-2

SP - 107

EP - 131

AB -
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.

LA - eng

KW - Stochastic differential inclusion; viable solution; tangential condition; Lipschitz property; stochastic differential inclusion

UR - http://eudml.org/doc/275927

ER -

## References

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- [2] J.P. Aubin, G. Da Prato, Stochastic Nagumo's viability theorem, Stochastic Analysis and Applications, 13 (1995), 1-11. Zbl0816.60053
- [3] N. Dunford, J.T. Schwartz, Linear Operators, Part I, Interscience Publisher Inc., New York 1957. Zbl0128.34803
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- [5] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, New York 1988. Zbl0638.60065
- [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. Zbl0796.60071
- [8] X.D.H. Truong, Existence of viable solutions of nonconvex-valued differential inclusions in Banach spaces, Portugalae Mathematica, 52 (1995), 241-250. Zbl0824.34021
- [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.

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