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Successive approximations of solutions to stochastic functional differential equations. (English) Zbl 0816.60055

The author considers the stochastic functional differential equation

dx(t)=f(t,x t )dt+g(t,x t )dw(t),t0,(*)

where x(t)=φ(t), tI 0 =[-r,0]. Applying the successive approximations method the author proves the local existence theorem and next he gives a sufficient condition for the global existence solution of (*).

60H10Stochastic ordinary differential equations
34F05ODE with randomness
[1]E. A. Coddington, N. Levinson,Theory of Ordinary Differential Equations, Mc Graw-Hill, New York, 1955.
[2]A. Friedman,Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, 1975.
[3]R. S. Lipster, A. N. Shiryayev,Statistics of Random Processes, Springer-Verlag, Berlin, 1977.
[4]S. E. A. Mohammed,Stochastic Functional Differential Equations, Pitman Publishing Program, Boston, 1984.
[5]A. E. Rodkina, On existence and uniqueness of solution of stochastic differential equation with hereditary,Stochastics 12 (1984), 187–200.
[6]T. Taniguchi, Successive approximations to solutions of stochastic differential equations,J. Diff. Eq. 96 (1992), 152–169. · Zbl 0744.34052 · doi:10.1016/0022-0396(92)90148-G
[7]E. F. Tsarkov,Random perturbations of functional-differential equations (Russian), ”Zinatne” Riga, 1989.
[8]T. Yamada, On the successive approximations of solutions of stochastic differential equations,J. Math. Kyoto Univ. 21 (1981), 501–515.