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

Abstract

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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.

How to cite

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Benoit 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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  1. [1] J.P. Aubin, G. Da Prato, Stochastic viability and invariance, Annali Scuola Normale di Pisa, 27 (1990), 595-614. Zbl0741.60046
  2. [2] J.P. Aubin, G. Da Prato, Stochastic Nagumo's viability theorem, Stochastic Analysis and Applications, 13 (1995), 1-11. Zbl0816.60053
  3. [3] N. Dunford, J.T. Schwartz, Linear Operators, Part I, Interscience Publisher Inc., New York 1957. Zbl0128.34803
  4. [4] S. Gautier, L.Thibault, Viability for constrained stochastic differential equations, Differential and Integral Equations 6 (6) (1993), 1395-1414. Zbl0780.93085
  5. [5] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, New York 1988. Zbl0638.60065
  6. [6] M. Kisielewicz, Viability theorem for stochastic inclusions, Discussiones Mathematicae - Differential Inclusions 15 (1995), 61-74. 
  7. [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. [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. [9] X.D.H. Truong, An existence result for nonconvex viability problem in Banach spaces, Preprint N.16 (1996), University of Pau, France. 
  10. [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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