Can someone help me with this exercise ?
Suppose that $(M_t)_{t \in [0,1)}$ is a bounded continuous martingale with finite variation. Prove that $ M_t^2 = M_0^2 + 2 \int_0^t M_s dM_s$, where the final integral is almost surely well-defined. (Hint: Cite here known results about one-dimensional functions of bounded variation.) Deduce that M is almost surely constant.