Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and bounded Ito process (w.r.t. $W$) with bounded drift and volatility coefficients (I could also live with more restrictions). I want to prove that \begin{align*} A := \mathbb E\bigg( \exp\bigg( \sup_{t\in[0;1]}\int^t_0 \sigma_s \mathrm dW_s\bigg) \bigg) < \infty. \end{align*}
If $\sigma$ was constant, this would be obvious since the distribution of the running maximum of the Brownian motion is known and the above expectation can be computed in closed integral form.
Question: Any ideas how to bound $A$ in a more general case than "$\sigma$ constant"? Or can such a result be found in the literature? Thank you!
Since this question got no answers after a few days, I also posted it on mathoverflow.
What I have tried: As an easier exercise, first I tried to prove the following: \begin{align*} B := \mathbb E\bigg( \exp\bigg( \int^1_0 \sigma_s \mathrm dW_s\bigg) \bigg) < \infty. \end{align*} This is also easy since the stochastic exponential is a martingale and then \begin{align*} 1 =& \mathbb E\bigg( \exp\bigg( \int^1_0 \sigma_s \mathrm dW_s - \frac 1 2 \int^1_0 (\sigma_s)^2 \mathrm ds \bigg) \bigg) \\\ge& \mathbb E\bigg( \exp\bigg( \int^1_0 \sigma_s \mathrm dW_s \bigg) \bigg) e^{- \frac 1 2 \Vert \sigma\Vert^2} \\=& B e^{- \frac 1 2 \Vert \sigma\Vert^2}. \end{align*}
Moreover, I thought we could $L^2$-approximate the Ito integral by \begin{align*} \sum_{k=1}^n \sigma_{(k-1)/n} ( W_{k/n} - W_{(k-1)/n} ), \end{align*} but I don't know how to approximate the supremum and anyway we cannot pull $L^2$-convergence into the $\exp$ function because $\exp$ increases faster than $\operatorname{id}^2$.