on the solution set of the nonlocal problem of it^o stochastic differential equation
Maysa Elgendy,
Published in International Journal of Advanced Research in Mathematics and Applications
ISSN: 2350-028X Impact Factor:1.2 Volume:1 Issue:2 Year: 05 July,2016 Pages:86-98
Abstract
In this paper we are concerned with two It^o problems of stochastic differential equation with nonlocal condition, the solutions are represented as stochastic integral equations that contain It^o integral or in a special case mean square Riemann-Steltjes integral. We study the existence of at least mean square continuous solution for these types. The existence of the maximal and minimal solutions will be proved.
Kewords
It^o integral, mean square Riemann-Steltjes integral, Brownian motion, random Caratheodory function, stochastic Lebesgue dominated convergence theor
Reference
[1] L. Arnold, Stochastic differential equations :theory and applications, A Wiley- Interscience Publication Copyright by J. Wiley and Sons, New York, 1974. [2] A. T. Bharucha-Teid, fixed point theorems in probabilistic analysis, Bulletin of the American Mathematical Society , 82, 5, 1976. [3] A. Boucherif, A first-order differential inclusions with nonlocal initial conditions, Applied Mathematics Letters, 15, 2002, PP. 409-414. [4] A. Boucherif and Radu Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4, 2, 2003, PP. 205-212. [5] L.Byszewski and V.Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Applicable analysis, 40, 1991, PP. 11-19. [6] N. Dunford, j.T. Schwartz, Linear operators,Interscience, Wiley, New York, 1958. [7] A.M. A. El-Sayed, R. O. Abd El-Rahman and M. El-Gendy, Uniformly stable solution of a nonlocal problem of coupled system of differential equations, Ele-Math-Differential Equattions and applications,5,3, 2013, PP. 355-365. [8] A.M. A. El-Sayed, R. O. Abd El-Rahman and M. El-Gendy, Existence of solution of a coupled system of differential equation with nonlocal conditions, Malaya Journal Of Matematik,2(4), 2014, PP. 345-351. [9] M. A. El-Tawil, On the application of mean square calculus for solving random differ- ential equations, Electronic Journal of Mathematical Analysis and Applications, 1(2), 2013, PP. 202-211. [10] D. Isaacson, Stochastic integrals and derivatives, The Annals of Mathematical Statis- tics,40(5), 1969, PP. 1610-1616. [11] K.Ito, On stochastic differential equations, Mem. A.M.S., 4, 1951, PP. 1-51. [12] S. Itoh, Random fixed point theorems with an application to random differential equa- tions in Banach spaces, Journal Of Mathematical Analysis And Applications, 67, 1979, PP. 261-273. [13] J. P. Noonan and H. M. Polchlopek, An Arzela-Ascoli type theorem for random func- tions, Internat. J. Math. and Math. Sci., 14(4), 1991, PP. 789-796. [14] A. Pisztora, Probability theory , New york university, mathematics department, spring 2008. [15] A. N. V. Rao and C. P. Tsokos, On a class of stochastic functional integral equation, Colloquium Mathematicum, 35, 1976, pp. 141146. [16] A. Shapiro, D. Dentcheva, and A. Ruszczynski Lectures on stochastic programming, modeling and theory, second edition, amazon.com google books,2014. [17] T. T. Soong, Random differential equations in science and engineering, Mathematics in Science and Engineering, 103, 1973.
