Given the SDE: $dX_{t}=\sqrt{X_{t}}dW_{t},$ $\ X_{0}=1$ , where $W_{t}$ is a 1-d Brownian motion.
I was told that this SDE has a unique strong solution, but I don't know how to construct it. I know that this SDE has strong uniqueness, therefore I only need to construct a weak solution. I'm guessing we need to first consider a weak solution up to an explosion time (i.e. the hitting time of $X_{t}$ to the level $0$), but how to show such a weak solution exists?
Thank you for your help!