Let $B$ be a Brownian motion, show that $X_t = e^{B_t}$ is a submartingale and find the Doob-Meyer decomposition of $X$.
Proving submartingality is straight forward. For Doob Decomposition. I looked into the discrete time predictable process and choosing ${t^n_i}=i2^{-n},\space i=0,1,...,2^n$$$\begin{align*} A^n_1=\sum_iE[e^{B_{t^n_{i+1}}}-e^{B_{t^n_i}} \mid F_{t^n_i}] &=\sum_iE[e^{B_{t^n_{i+1}}} \mid F_{t^n_i}]-e^{B_{t^n_i}} \\ &=\sum_i \left(e^{B_{t^n_{i}}}e^{{t^n_{i+1}}-{t^n_{i}}\over2}-e^{B_{t^n_i}}\right) \\ &=\sum_i{e^{B_{t^n_{i}}}(e^{{{t^n_{i+1}}}-{t^n_{i}}\over2}-1)}\end{align*}$$ Any hints on how to proceed?