For a semimartingale $X$, we want to solve the SDE $dZ_t=Z_{t-}dX_t$. I was able to prove that $ Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t^c)}\prod_{0
is the solution if I can show that $Z_s=Z_{s-}(1+\Delta X_s)$ and $Z_{s-}\Delta K_s=Z_{s-}\Delta X_s$, where $K_s:=X_t-\frac{1}{2}\langle X\rangle_t^c$. Can someone tell me, why these equations are true?
Thanks in advance!