1. On the internet, it is suggested that $ X_t=\left(\int_0^t X_s \;ds\right)\;dW_{t} $ is a martingale, but not a Markov process. I understand that the process $ I_t(C)=\int_0^t C_s \; dW_s$ is a martingale, but I don't see $X_t$ as an Ito's integral. In order to show it is a martingale but not a Markov process, I want to show that $ E(X_t\mid\mathcal{F}_{t-1})=X_{t-1}\neq E(X_t\mid\sigma(X_{t-1})). $ I don't know how to take the expecation in this case.
Other counterexamples available online might be wrong. (Or, I could be wrong.)
2. Invalid counterexample: Let $e(t)$ be a random variable which is not independent of $y(0)$ and $ y(t)=y(t-1)+e(t)-E{[}e(t)|F(t-1){]}. $ Then y(t) is a martingale but it is not a Markov process.
I will show that it is both a martingale and a Markov process. Define $X_{t}$ as $ X_{t}=X_{t-1}+e_{t}-E\left(e_{t}|\mathcal{F}_{t-1}\right) $
Then, $\{X_{t}\}$ is a martingale because $ E(X_{t}|\mathcal{F}_{t-1}) = E\left(X_{t-1}+e_{t}-E\left(e_{t}|\mathcal{F}_{t-1}\right)\bigg|\mathcal{F}_{t-1}\right) = E(X_{t-1}|\mathcal{F}_{t-1})+E\left(e_{t}|\mathcal{F}_{t-1}\right)-E\left(e_{t}|\mathcal{F}_{t-1}\right) = E(X_{t-1}|\mathcal{F}_{t-1}) = X_{t-1} $
However, it is also a Markov process. $ E(X_{t}|\sigma(X_{t-1})) = E\left(X_{t-1}+e_{t}-E\left(e_{t}|\mathcal{F}_{t-1}\right)\bigg|\sigma(X_{t-1})\right) = E(X_{t-1}|\sigma(X_{t-1}))+E\left(e_{t}|\sigma(X_{t-1})\right)-E\left(E\left(e_{t}|\mathcal{F}_{t-1}\right)\bigg|\sigma(X_{t-1})\right) $
Since $\sigma(X_{t-1})\subset\mathcal{F}_{t-1}$, by the towering property, $ E\left(\left(e_{t}|\mathcal{F}_{t-1}\right)\bigg|\sigma(X_{t-1})\right)=E\left(e_{t}|\sigma(X_{t-1})\right). $
Thus $E(X_{t}|\sigma(X_{t-1})) = E(X_{t-1}|\sigma(X_{t-1}))+E\left(e_{t}|\sigma(X_{t-1})\right)-E\left(e_{t}|\sigma(X_{t-1})\right) = E(X_{t-1}|\sigma(X_{t-1}) $
Since $X_{t-1}$ is $\sigma(X_{t-1})$-measurable, $E(X_{t-1}|\sigma(X_{t-1}))=X_{t-1}$. Thus $ E(X_{t}|\sigma(X_{t-1}))=X_{t-1}=E(X_{t}|\mathcal{F}_{t-1}) $ Hence $\{X_{t}\}$ is also a Markov process.
3. Invalid counterexample from the textbook answer key. Alison Etheridge's textbook gives an example: $ S_{n+1}=S_{n}+\xi_{n+1} $ where $\xi_{n}\in\{-1,1\}$ and $\{\xi_{n}\}_{n\geq0}$ are iid. Then, $ P\left(S_{n+1}=k+1\bigg|S_{n}=k\right)=p \text{and }P\left(S_{n+1}=k-1\bigg|S_{n}=k\right)=1-p. $ The author claims that $\left\{ S_{n}\right\} $ is a Markov process.
Then in the homework answer key, someone gives $\left\{ Z_{n}\right\}$ as a counterexample of a martingale not being a Markov process: $ \displaystyle{Z_{n}=\sum_{i=1}^{n}\xi_{i}} $ where $P(\xi_{i}=1)=\frac{1}{2}=P(\xi_{i}=-1)$.
Obviously, $Z_{n+1}=Z_{n}+\xi_{n+1}.$ So what's the difference between this counterexample and the example? Moreover, given $P(\xi_{i}=1)=\frac{1}{2}=P(\xi_{i}=-1)$, I think is both $\left\{ Z_{n}\right\}$ is both a martingale and a Markov process.