I quote a paper from Delbaen and Shirakawa (2002). I will write in *italics* my observations/questions.

Starting from a stochastic differential equation of the form:

$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\tag{1}$$ with $\left\{W_t\right\}_{t\geq0}$ a standard Wiener process in the filtered probability space $\left(\Omega,\mathcal{F},\left\{\mathcal{F}_n\right\},\mathbb{P}\right)$. We assume $\alpha,\beta>0$ and $r_m<r_{\mu}<r_M$, which guarantee the existence of stationary distribution.

(1. Why do the assumption $\alpha,\beta>0$ and $r_m<r_{\mu}<r_M$ guarantee stationarity of distribution? What is exactly meant here?)

$\alpha$ represents the speed of reversion to the longrun mean $r_{\mu}$.

Then, for diffusion of $(1)$, set $\sigma(x)=\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}$. For any $x,y\in[r_m,r_M]$, it holds that: $$|\sigma^2(x)-\sigma^2(y)|=\beta|r_m+r_M-(x+y)||x-y|\leq\beta(r_M-r_m)|x-y|\tag{2}$$ This means that the diffusion coefficient function $\sigma(x)$ is Holder $\frac{1}{2}$ continous.

(2. Why does $(2)$ translate into the fact that $\sigma(x)$ is Holder $\frac{1}{2}$ continous? Doesn't $(2)$ denote instead that $\sigma^2(x)$ is Holder continous (with coefficient $1$)?)

Since $\mu(x)=\alpha(r_{\mu}-x)$ is Lipschitz continuous, the pathwise uniqueness of the stochastic differential equation is guaranteed by the general uniqueness theorem.

(3. Could you please give me some reference for such a theorem? In this specific case, does this theorem rely on Lipschitz condition on $\mu(x)$ combined with Holder continuity of $\sigma(x)$? If so, why do such two conditions guarantee the uniqueness of the path of SDE $(1)$?)