Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $\mathbb{R}$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e.

$$\sup_{t,\omega}|H_t(\omega)|<+\infty$$

My question is whether we could show the stochastic integral

$$\int_0^{\cdot}H_udM_u$$

is a martingale? I stronly believe that the answer is no, but I can't find a counterexample. Does someone know the result? Many thanks for the reply!