On a transformation of systems of stochastic differential equations

For a diffusion Markov process defined by the Ito equations (1), an $\mathfrak{F}_t$-measurable transformation defined by (3) or (4) with $G(z,t)$ and $F(x,t)$ satisfying (2) and (6) respectively is considered. The process $(z(t), y(t))$ where $z(t)=F(x(t), t)$ with $(x(t), y(t))$ defined by (1) is shown to satisfy the system (5).