Stochastic process which is not a martingale

Question

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

Answer 1

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 $s,t$).

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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.