Given a deterministic function $h\in L^2([0,T]; \mathbb{R})$, we can define the associated exponential martingale \begin{align} M_t = \exp\left[\int_{0}^{t} h_s \,dB_s - \frac{1}{2}\int_{0}^{t} h_s^2\,ds\right], \quad t\in [0,T] \end{align} with the "kernel" $h$. Here $B$ is a standard Brownian motion. By Ito's formula, we obtain $dM_t=M_t h_t dB_t$.

I was wondering, if I choose a sequence of functions $h^n\in L^2([0,T]; \mathbb{R})$, such that $h^n\to h$ in $L^2([0,T]; \mathbb{R})$, with some additional assumptions for $h^n$ or $h$, is it possible to get the following convergence result in $L^2(\Omega; \mathbb{R})$ $$M_T^n\to M_T,\quad n\to \infty$$ where $M_T^n$ is the exponential martingale associated to $h_n$.

I get this question from the Ito representation theorem, which basically results from the density of exponential martingales associated to piecewise constant functions. However, piecewise constant functions on $[0,T]$ is uncountable, hence I try to use a countably dense subset of $L^2([0,T]; \mathbb{R})$ (say some good polynomials) to approximate such functions.

I asked the same question on MSE two weeks ago but haven't received any answer till now.