I'm trying to find results that estimate the stochastic exponential $\mathcal{E}(M)$ of a continuous local martingale $M = (M_t)_{0 \leq t \leq T}$ starting at zero, where $T \in (0,\infty)$. By definition, $\mathcal{E}(M)$ is the unique solution to the SDE $$ Y_t = Y_0 + \int_0^t Y_s d M_s, \quad 0 \leq t \leq T, $$ and can explicitely written as $$ \mathcal{E}(M)_t = \exp \left( M_t - \frac{\langle M \rangle_t}{2} \right), \quad 0 \leq t \leq T, $$ and also by definition it is a continuous local martingale.
The particular case I'm looking for is where $M$ itself is a Brownian stochastic integral, that is, $M_t = \int_0^t H_s d W_s$ for a progressive process $H$ and a Brownian motion $W$. Then $$ \mathcal{E}(M)_t = \exp \left( \int_0^t H_s d W_s - \frac{1}{2}\int_0^t H^2_s ds \right) $$ Can we say something about the boundedness of the stochastic integral $\mathcal{E}(M)$ if we impose conditions on the process $H$? For example, what if $H$ is bounded? Is the stochastic exponential then also bounded?