Approximate an exponential martingale through its kernel

Question

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 $h_n$.

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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.

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

Answer 1

The OP seems to be an $\epsilon$ or so away from answering the question. Let me try to plug the gap, by introducing the approximating SDEs that $M_t^n$ satisfies: $$ d M_t^n = h_t^n M_t^n d B_t \;, \qquad M_0^n = 1 \;, \tag{a} $$ which we will compare to $$ d M_t = h_t M_t d B_t \;, \qquad M_0 = 1 \;. \tag{b} $$ As long as $h$ and $\{ h_n \}$ are continuous and bounded on $[0,T]$, then the coefficients of these SDEs fulfill standard global Lipschitz and linear growth conditions. Thus, these SDEs have unique strong solutions whose higher moments are nicely bounded on finite time intervals; to read more about this see, e.g., Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.

Moreover, by Itô isometry \begin{align*} E | M_t - M_t^n |^2 &= E \left( \int_0^t (M_s h_s - M_s^n h_s^n) d Bs \right)^2 \\ &= E \int_0^t |M_s h_s - M_s^n h_s^n|^2 ds \\ &\le E \int_0^t |M_s h_s - M_s h_s^n + M_s h_s^n - M_s^n h_s^n|^2 ds \\ &\le 2 \int_0^t |h_s-h_s^n |^2 E M_s^2 ds + 2 \int_0^t |h_s^n|^2 E |M_s - M_s^n |^2 ds \\ &\le C(t) \exp\left( 2\int_0^t |h_s^n|^2 ds \right) \int_0^t |h_s - h_s^n|^2 ds \end{align*} where in the last step we used Gronwall's inequality and a bound on the second moment of $M_t$ over $[0,t]$ which we lumped into a positive constant $C(t)>0$.

Note that this last inequality allows one to use convergence of the sequence of (deterministic) functions $\{h_n\}$ in $L^2([0,t]; \mathbb{R})$ to obtain mean-squared convergence of $M_t^n$ to $M_t$, as requested by the OP.