Consider the ODE

\[ \dot x_t = x_t(1-x_t)\]

and the SDE

\[dX_t = X_t(1-X_t) dt + \sqrt{X_t(1-X_t)} dW_t\]

1. Argue that \(x_t\) can not leave the interval \([0,1]\) if \( x_0 \in (0,1)\).
2. What is the behavior of \(x_t\) as \(t \rightarrow\infty\) if if \( x _0\in (0,1)\) ?
3. Can the diffusion \(X_t\) exit the interval \(  (0,1) \) ? Prove your claims.
4. What do you think happens to \(X_t\) as \(t \rightarrow \infty\) ? Argue as best you can to support your claim.