Back Ground
I read books "Diffusion Models" written by Okanohara Daisuke. I cannot understand bleow problem.
Please tell me how to prove this problem.
Problem
We consider below stochastic differential euqation. \begin{equation} dx=f(t)xdt+g(t)dw \end{equation}
Then we get conditional probability distribution $p(x(t)|x(0))$. \begin{eqnarray} p(x(t)|x(0))&=&\mathcal{N}(s(t)x(0),s(t)^2\sigma(t)^2I)\\ s(t)&=&\exp\left(\int_{0}^{t}f(\xi)d\xi\right)\\ \sigma(t)&=&\sqrt{\int_{0}^{t}\dfrac{g(\xi)^2}{s(\xi)^2}d\xi} \end{eqnarray}