I have a question about super-martingale and martingale convergence theorem.
Fix $T>0,\lambda>0$. For a continuous local martingale $X=(X)_{t \in [0,T]}$, we can define
\begin{align*} \mathcal{E}^{\lambda}(X)_{t}=\exp \left\{ \lambda X_{t}-\frac{\lambda^{2}}{2} \langle X,X\rangle_{t} \right\} \end{align*}
By Itô's formula, $\mathcal{E}^{\lambda}(X)$ is a local martingale. Since $\mathcal{E}^{\lambda}(X)$ is positive, futhermore, we can show that $\mathcal{E}^{\lambda}(X)$ is super-martingale.
My question
When $X_{0}=0$, can we show that $\displaystyle \mathcal{E}^{\lambda}(X)_{t} \underset{t \to0}\longrightarrow 1$ in $L^{1}(\Omega)$ ?
My attempt
It is clear that $\lim_{t \to 0}\mathcal{E}^{\lambda}(X)_{t}= 1$ $P$-a.s. . If $\mathcal{E}^{\lambda}(X)$ is martinagle, $(\mathcal{E}^{\lambda}(X)_{t})_{t \in [0,T]}$ is uniformly integrable. Therefore we can get $\displaystyle \mathcal{E}^{\lambda}(X)_{t} \underset{t \to0}\longrightarrow 1$ in $L^{1}(\Omega)$.
Can we show that $\displaystyle \mathcal{E}^{\lambda}(X)_{t} \underset{t \to0}\longrightarrow 1$ in $L^{1}(\Omega)$ if $\mathcal{E}^{\lambda}(X)$ is super-martingale ?
Thank you in advance.