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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

$dx\left(t\right)=f\left(t,{x}_{t}\right)dt+g\left(t,{x}_{t}\right)dw\left(t\right),\phantom{\rule{1.em}{0ex}}t\ge 0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $x\left(t\right)=\phi \left(t\right)$, $t\in {I}_{0}=\left[-r,0\right]$. 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 $\left(*\right)$.

##### MSC:
 60H10 Stochastic ordinary differential equations 34F05 ODE with randomness
##### References:
 [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.