Prove that if (theta) is a bounded function, then the process (left(mathcal{E}(theta star W)_{t}, t leq T ight)) is a u.i. martingale. [exp left(int_{0}^{t} theta_{s}

Prove that if \(\theta\) is a bounded function, then the process \(\left(\mathcal{E}(\theta \star W)_{t}, t \leq T\right)\) is a u.i. martingale.

\[\exp \left(\int_{0}^{t} \theta_{s} d W_{s}-\frac{1}{2} \int_{0}^{t} \theta_{s}^{2} d s\right) \leq \exp \left(\sup _{t \leq T} \int_{0}^{t} \theta_{s} d W_{s}\right)=\exp \widehat{\beta}_{\int_{0}^{T} \theta_{s}^{2} d s}\]

with \(\widehat{\beta}_{t}=\sup _{u \leq t} \beta_{u}\) where \(\beta\) is a BM.