Let $M$ martingale limited, continuous, integrable with a square. Prove that $$\int_0^t M_sdM_s=\frac 12 (M_t^2 - \langle M \rangle _t)$$
This theorem is proven on the forum and in many books but for Brownian motion. Unfortunately, I don't know how to prove it in the general case - for a martingale with the given in task properties.
Important note - Ito's formula has not yet been derived during the lecture, so we are not allowed to use it.