I am trying to form martingales from various combinations of $$A_t=\int_0^tX_s\,\text{d}M_s,$$ $$B_t=\int_0^tY_s\,\text{d}N_s,$$ $$C_t=\int_0^tX_s\,\text{d}N_s,$$ $$\text{and }D_t=\int_0^tY_s\,\text{d}M_s,$$ with $M_s$ and $N_s$ being continuous, square integrable martingales (I presume true, not just local) and the integrals of $X^2_s$ and $Y^2_s$ with respect to the quadratic variations of $M_s$ and $N_s$ are finite.
So far what I have seen is Brownian motion as the integrator. However, for general martingales as the integrator, I am not sure how to prove various combinations are martingales, or construct martingales from them. My assumption (extrapolating from what I know should be martingales when in the case of a BM) is $$A_tD_t-\langle A,D\rangle_t,$$ $$A_tB_t-\langle A,B\rangle_t,$$ $$(A_t+C_t)(B_t+D_t)-2\int_0^tX_sY_s\,\text{d}\langle M+N\rangle_s,$$ $$\text{and }A_tB_t+C_tD_t-2\int_0^tX_sY_s\,\text{d}\langle M,N\rangle_s.$$ I am not exactly sure how to begin with my proofs, do I have to do it in the usual way of decomposing them into a summation and setting the limit of the supremum of the partition to zero (like when we do for the Ito integral)? Really quite clueless. Cheers!