Suppose $ X = \left ( X_t \right )_{t\geq 0} $ is a stochastic process.
1) Under which conditions on $ X $ does \begin{equation} \lim_{t\rightarrow 0} \mathbb{E}\lvert X_t \rvert = \mathbb{E}\lvert X_0 \rvert \end{equation} hold?
2) Under which conditions on $ X $ does \begin{equation} \lim_{t\rightarrow 0} \mathbb{E}\lvert e^{X_t} \rvert = \mathbb{E}\lvert e^{X_0} \rvert \end{equation} hold?
Possible conditions I thought of are $ X $ being cadlag or $ X $ being a Levy process.
Thanks!