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Successive approximations to solutions of stochastic differential equations. (English) Zbl 0744.34052

This paper studies the Itô stochastic differential equation

$dX\left(t\right)=f\left(t,X\left(t\right)\right)+g\left(t,X\left(t\right)\right)dw\left(t\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

with initial condition $X\left(0\right)={X}_{0}$ a.e., where the Brownian motion $w\left(t\right)$ and the random variable ${X}_{0}$ are independent and $E\left(|{X}_{0}{|}^{2}\right)<\infty$. Under somewhat less restrictive conditions than previously used, the uniform convergence of the successive approximations

${X}_{n}\left(t\right)={X}_{0}+{\int }_{{t}_{0}}^{t}f\left(\tau ,{X}_{n-1}\left(\tau \right)\right){d}^{\tau }+{\int }_{{t}_{0}}^{t}g\left(\tau ,{X}_{n+1}\left(\tau \right)\right)dw\left(\tau \right)$

to the solution $X\left(t\right)$ of initial value problem (1) is proved which also establishes the local existence and uniqueness of this solution. Then the paper concludes by proving global uniform convergence of the successive approximations and global existence-uniqueness of the solution of IVP (1) under appropriately amended conditions.

MSC:
 34F05 ODE with randomness 34A45 Theoretical approximation of solutions of ODE 60H10 Stochastic ordinary differential equations