# BOUNDEDNESS AND CONTINUITY OF SOLUTIONS FOR STOCHASTIC DIFFERENTIAL INCLUSIONS ON INFINITE DIMENSIONAL SPACE

• Yun, Yong-Sik (DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCES CHEJU NATIONAL UNIVERSITY) ;
• Ryu, Sang-Uk (DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCES CHEJU NATIONAL UNIVERSITY)
• Published : 2007.11.30
• 62 4

#### Abstract

For the stochastic differential inclusion on infinite dimensional space of the form $dX_t{\in}\sigma(X_t)dW_t+b(X_t)dt$, where ${\sigma}$, b are set-valued maps, W is an infinite dimensional Hilbert space valued Q-Wiener process, we prove the boundedness and continuity of solutions under the assumption that ${\sigma}$ and b are closed convex set-valued satisfying the Lipschitz property using approximation.

#### Keywords

stochastic differential inclusion;Wiener process

#### References

1. N. U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stochastic Anal. Appl. 12 (1994), no. 1, 1-10 https://doi.org/10.1080/07362999408809334
2. A. A. Levakov, Asymptotic behavior of solutions of stochastic differential inclusions, Differ. Uravn. 34 (1998), no. 2, 204-210
3. G. D. Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992
4. B. Truong-Van and X. D. H. Truong, Existence results for viability problem associated to nonconvex stochastic differentiable inclusions, Stochastic Anal. Appl. 17 (1999), no. 4, 667-685 https://doi.org/10.1080/07362999908809628
5. Y. S. Yun, On the estimation of approximate solution for SDI, Korean Ann. Math. 20 (2003), 63-69
6. Y. S. Yun and I. Shigekawa, The existence of solutions for stochastic differential inclu- sion, Far East J. Math. Sci. (FJMS) 7 (2002), no. 2, 205-212
7. J. P. Aubin and G. D. Prato, The viability theorem for stochastic differential inclusions, Stochastic Anal. Appl. 16 (1998), no. 1, 1-15 https://doi.org/10.1080/07362999808809512
8. B. Truong-Van and X. D. H. Truong, Existence of viable solutions for a nonconvex stochastic differential inclusion, Discuss. Math. Differential Incl. 17 (1997), no. 1-2, 107-131