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Given a probability space, we say that $(X_t)_{t \geq 0}$ is Markov w.r.t its own filtration $(\mathcal F_t)$ if for all $s<t$,

$$ P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_s).$$

Constructing continuous Markov processes that are Gaussian is easy as Markovianity is captured by the covariance function in the Gaussian case. The canonical example of a continuous Markov process that is Gaussian is Brownian motion.

Now what about the non-Gaussian case ? What is the canonical example of a continuous Markov process that is not Gaussian ? I actually cannot think about a single process that is continuous, Markov and not Gaussian. I feel like there should be a way of playing with SDEs so that the solution is Markov and not Gaussian, but I hope there is an "explicit" example.

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    $\begingroup$ Some quick examples that comes to my mind are the geometric Brownian motion and the Bessel process. $\endgroup$ Commented Nov 4, 2021 at 2:33
  • $\begingroup$ @SangchulLee Of course, the geometric BM! I did not know about Bessel processes, this is very helpful. $\endgroup$
    – W. Volante
    Commented Nov 4, 2021 at 3:08

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Let $X_t$ be your favorite continuous Gaussian Markov process, and let $f$ be a continuous bijection of $\Bbb R$ onto $(0,1)$. Then $Y_t:=f(X_t)$ is continuous and Markov, but not Gaussian.

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Trivial example: let $Z = \pm 1$ with probability 1/2, and $X_t = tZ$. Note that $\mathcal{F}_s = \sigma(X_s) = \sigma(Z)$ for every $s > 0$.

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