Stochastic process which is not a martingale What is an example of a stochastic process, $X_t$, which is not a martingale and has $\mathbb{E}[X_t]=t$?

There are no martingales whose expectations is $\mathbb{E}[X_{t}]=t$ since martingales have constant expectation (i.e., $\mathbb{E}[X_{t}]=\mathbb{E}[X_{s}]$ for all $s,t$).

The above claim is easy to show. Recall that the defining property of martingales is that
\begin{align}
\mathbb{E}[X_{t}|\mathcal{F}_s]=X_s, \quad \forall s,t\;\text{ s.t. }\;s<t
\end{align}
Taking expectations of both sides of the above equation we obtain
\begin{align}
&\mathbb{E}[\mathbb{E}[X_{t}|\mathcal{F}_s]]=\mathbb{E}[X_s],
\end{align}
which, by the tower property of conditional expectation, simplifies to
\begin{align}
\mathbb{E}[X_{t}]=\mathbb{E}[X_s]
\end{align}
Thus, it follows that if $(X_t)$ is a martingale then its expectations are constant.
