平面上Volterra型随机微分方程的弱解

THE WEAK SOLUTION OF VOLTERRA STOCHASTIC DIFFERENTIAL EQUATION IN THE PLANE

设 B 是(?)上的 Brown 运动,考虑平面上 Volterra—It(?)型随机微分方程(Ⅰ)X_(?)=(?)+(?)a(z,ξ,X_ξ)dξ+∫_(R_z)β(z,ξ,X_(?))dB_(?) z∈R_+~2其中(?)是两参数连续过程,满足:对(?)T>0,(?),则当α(z,ξ,x),β(z,ξ,x)连续,且关于 z 满足 Lip 条件,关于 x 满足增长性条件时,本文用迟滞逼近方法证得方程(Ⅰ)弱解存在。

We consider Volterra—It(?) SDE in the plane:X_z=φ_z integral (R_z) α(z,ξ,X,)dξ+integral (R_z)β(z,ξ,X_ξ)dB_ξ z∈R_+~2where φ_z is a two—parameter continuous adapted process,fro (?)T>0 (?)Under the assumption that the coefficients a(z,ξ,x),β(z,ξ,x,)satisfya Lipschitz condition on z,and a growth condition on x,we prove the existe-nce of a weak solution by the method of delayed approximations.