For a given Weiner process W consider the 1-d interest rate model
\begin{align*} dr_t&=4(4-r_t)dt+\sqrt{|r_t-6||r_t-2|}dW_t, t\in[0,T] \\ r_0&=4. \end{align*}
Show that this equation has a unique solution and that with probability $1$ the solution lies in the interval $[2,6]$ with constant mean $4$.
I'm struggling to answer this question. I was given a hint to write down the equivalent equation for the centered process $x_t=r_t-4$ which I found to be $$dx_t=-4x_tdt+\sqrt{|x_t-2||x_t+2|}dW_t$$ but I'm not sure how I'm meant to use this?
For simplicity let us directly consider the equation for $x_t$. Uniqueness can be easily proved thanks to classic theorems on this topic. (For instance, check here). In your case, it will be enough to observe that:
$$ |4x| + \sqrt{|x-2| |x+2|} \leq 5 (1+|x|) $$ $$ |4x-4y| + \left| \sqrt{|x-2| |x+2|} - \sqrt{|y-2| |y+2|} \right| \leq 5 |x-y|. $$
Next, to show boundaries, note that $x_t$ starts in $0$ and the term multiplying $dW_t$ in the SDE is positive and continuous. When $x_t$ tends towards $\pm2$, the noise will be lower and lower, eventually $0$ when $x_t=\pm2$. Hence, we are left with the drift term: e.g. when $x_t=2$, the drift will be $-8dt$, pushing $x_t$ lower, towards $0$. Similarly, when $x_t=-2$, then $x_t$ will be increased towards $0$. Hence, with probability $1$, $x_t$ cannot leave the set $[-2,+2]$.