Let $H$ be predictable and bounded. According to my lecture notes, for Brownian motion $W$, $\int HdW$ is a martingale.
Question
Is it true that for any $M\in\mathcal{H}_{0,\mathrm{loc}}^{2}:=\{\text{local martingales starting at zero s.t.} \sup_{t\ge0}\|H_t\|_{L^2(P)}<\infty\}$: $\int HdM$ is a martingale?
My approach
By definition, $\int HdM\in\mathcal{H}_{0,\mathrm{loc}}^{2}$ with, say, localizing sequence $(\tau_n)_{n\in\mathbb{N}}$. So we only need to show $E\big[\sup_{n\in\mathbb{N}}|\int_0^t H_udM^{\tau_n}_u|\big]<\infty$ for all $t\ge0$, and for this, it is sufficient to show $E\Big[\big(\int_0^t H_udM_u\big)^2\Big]<\infty$. But $$E\Big[\big(\int_0^t H_udM_u\big)^2\Big]=E\Big[\int_0^t H_u^2d[M]_u\Big]\le\|H\|_\infty^2 E\big[[M]_t\big]$$ and we have $E\big[[M]_t\big]<\infty$ iff $M\mathbb{1}_{[0;t]}\in\mathcal{H}_{0}^{2}$. However, I do not know what a good sufficient condition for $M\mathbb{1}_{[0;t]}\in\mathcal{H}_{0}^{2}$ could be.
By the Burkholder-Davis-Gundy inequalities, $E\big[[M]_t\big]<\infty$ iff $E\big[\big(\sup_{0\le s\le t}|M_s|\big)^2\big]<\infty$. If $M$ is a true martingale with $M_t\in L^2(P)$, Doob's inequality implies that this is finite.
Hence, for any martingale $M\in\mathcal{H}^2_{0,\mathrm{loc}}$ and predictable bounded $H$, $\int HdW$ is a martingale.