I've been tasked to solve the following SDE:
$dX_t = dt + 2\sqrt{X_t}dW_t, \ \ \ t \in \mathbb{R}_+$
where $W_t$ is a standard Brownian motion and $X_0=x$.
I need a closed form solution.
What is the best approach?
Is it to write in in "integral" form: $X_t = x + t + 2\int_0^t\sqrt{X_u}dW_u$ and solve the stochastic integral term? If so, I can't find a function $f$ to use in Ito's lemma so that I get a good-enough solution.
Any advice would be appreciated as I am pretty much a bigger to stochastic calculus. Thanks
Edit: Note: I am allowed to assume X stays non-negative all the time