Let $X$ and $Y$ be martingales, $X_0 = Y_0 = 0$.
Can the stochastic exponential $\mathcal{E}(X+Y+[X,Y]) _t$ be written on the form of $\mathcal{E}(X+Y)_t \exp (C_t)$ for some sort of "compensator" $C_t$? And if yes and not obvious what properties will it have?
The motivation stems from if $X,Y$ are continuous then $$ \mathcal{E}(X+Y+[X,Y])_t = exp ( X_t+Y_t+[X,Y]_t - 1/2 [X+Y+[X,Y]]_t] ) = exp ( X_t+Y_t - 1/2 [X+Y] ) exp ( [X,Y]_t) = \mathcal{E}(X+Y)_texp ( [X,Y]_t) $$ Since $[X,Y]_t$ is continuous and of finite variation, so then simply $C_t = [X,Y]_t$.