I have a sequence of processes $\{X^N(t)\}_{t\in [0,T]}$, $N\in\mathbb N$ such that $X^N(t)=x+M^N(t)$, where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $<M^N>(t)$ such that $<M^N>(t)-\int_0^tX^N(s)(1-X^N(s))ds\xrightarrow[N\to+\infty]{}0$ in probability.

I know that the process $X^N(t)$ converges (I proved its tightness), I would like to conclude that $X^N(t)\to X(t)$ in distribution, where $X(t)$ is the solution of the following system of SDE $$dX(t)=\sqrt{X(t)(1-X(t)}dB(t)$$ $$X(0)=x$$ with $B(t)$ the classic Brownian motion.

Could someone suggest me a book in which I can find a theorem that could help me with the convergence of martingales?