Successive approximations to solutions of stochastic differential equations.

*(English)*Zbl 0744.34052This paper studies the Itô stochastic differential equation

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(\right|{X}_{0}{|}^{2})<\infty $. Under somewhat less restrictive conditions than previously used, the uniform convergence of the successive approximations

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

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.

Reviewer: M.D.Lax (Long Beach)

##### MSC:

34F05 | ODE with randomness |

34A45 | Theoretical approximation of solutions of ODE |

60H10 | Stochastic ordinary differential equations |