# Martingales limit theorems (reference)

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 $$(t)$$ such that $$(t)-\int_0^tX^N(s)(1-X^N(s))ds\xrightarrow[N\to+\infty]{}0$$ 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?