Let $\alpha$ be an adapted process cadlag (right continuous with left limits), such that $\int_{0}^{\infty}E[\alpha_s^2]ds < \infty$ and $B$ a brownian motion. We define $\mathcal{E}_t(M) = \exp \{ M_t - \frac{1}{2}[M_t]\}$ for all $t \ge 0$ where $M = \int \alpha_s dB_s$. Show that $\{\mathcal{E}_t(M); t\ge 0 \}$ is a continuous martingale bounded in $\mathcal{L}^2$
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