Let X X be a semi-martingale. Check that the solution of the S D E d S

Let $X$ be a semi-martingale. Check that the solution of the $SDEd{S}_t=S_{t^{-}}\left(b\right(t)dt+\sigma (t\left)d{X}_t\right)$ is

$$S_t=S_0\exp \left(U_t\right){\Pi }_t$$

where

$$\begin{array}{cc}{U}_t& ={\int }_0^t\sigma \left(s\right)d{X}_s+{\int }_0^{t}b\left(s\right)ds-\frac{1}{2}{\int }_0^t{\sigma }^2\left(s\right)d{⟨X^c⟩}_s\\ {\Pi }_t& =\prod _{0<s\le t}(1+\sigma (s\left)\Delta X_s\right)\exp (-\sigma (s\left)\Delta X_s\right).\end{array}$$