Stochastic process with Student's t-distribution

Question

I have encountered the following SDE $$dX_t=-X_tdt+\sqrt{1+X_t^2}dW_t.$$ Does anyone know a name by which the resulting process $X_t$ is commonly referred to? In the place I found it, it is claimed that $X_t$ is t-distributed for any $t\geq0$. It is also featured as an example on the following Mathematica documentation page: https://reference.wolfram.com/language/ref/ItoProcess.html

Answer 1

Leonenko and Šuvak call it Student diffusion process. Not too "common", but there's no other name I am aware of.

Answer 2

It's not far off from the Hyperbolic diffusion:

$$dX_t = \alpha \frac{X_t}{\sqrt{1+X_t^2}}dt + \sigma dW_t$$ [ cf. Parameter Estimation in Stochastic Diffential Equations, page 2 or example 2.6 on page 47]

The example the author gives is as follows

$$dX_t = -\theta \frac{X_t}{\sqrt{1+X_t^2}}dt + \sigma dW_t$$ where $\theta \in (\alpha,\beta), \alpha>0$.