It型随机微分方程弱解的存在性

ON THE EXISTENCE OF A WEAK SOLUTION OF AN IT STOCHASTIC DIFFERENTIAL EQUATION

在关于 It型随机微分方程解存在的经典定理中,对方程的系数加有线性增长条件.本文证明了将线性有界条件添上一些多重对数因子后,方程依然有弱解存在.

It is shown that the scalar stochastic differential equation has a weak solution under following conditions: (i) Scalar function a (s,x) and b (s,x) are continuous in x. (ii)|a(s,x)|+|b(s,x)|≤K(s)(x) (iii) Eg_(m+1)(ζ_0)<∞ Where K(s) is non-negative and locally square intograble,g_m(x) are given by following recursion formula g_0(x)=1+|x|,x∈R g_m(x)=1+1ng_(m+1),(x),m=1,2,…