Given a deterministic function $h\in L^2([0,T])$, 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" $h$. By Ito's formula, we obtain $dM_t=M_th(t)dB_t$.
I was wondering, if I choose a sequence of functions $h_n\in L^2([0,T])$, such that $h_n\to h$ in $L^2([0,T])$, with some additional assumptions for $h_n$ or $h$, is it possible to get the following convergence result in $L^2(\Omega)$ $$M_T^n\to M_T,\quad n\to \infty$$ where $M_T^n$ is the associated exponential martingale with $h_n$.
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 $L^2([0,T])$ (say some good polynomials) to approximate such functions.