# A continuous Markov process that is not Gaussian?

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,

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

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.

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

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.
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 $$s > 0$$.