Let $Z_1, \dots, Z_n$ be independent random variables and define $X_n = Z_1 + Z_2 + \dots Z_n$ where $E[Z_{n+1} | \mathfrak{F}_n] = 0$. Is $Y_n = e^{X_n}$ a martingale, submartingale or supermartingale?
To solve this I calculated the expected value but got stuck here:
\begin{equation}
\begin{aligned}
E[Y_{n+1}| \mathfrak{F_n}] = E[e^{X_{n+1}}| \mathfrak{F_n}] = E[e^{X_{n} + Z_{n+1}}| \mathfrak{F_n}] = Y_n E[e^{Z_{n+1}}|\mathfrak{F_n}]
\end{aligned}
\end{equation}
Best Answer
By Jensen's inequality we have $1=e^{E(Z_{n+1}|\mathcal F_n)} \leq E(e^{z_{n+1}}|\mathcal F_n)$. Hence, $E(Y_{n+1}|\mathcal F_n) =Y_n E(e^{z_{n+1}}|\mathcal F_n)\geq Y_n$. Thus $(Y_n)$ is a submartingale.