Find closed-form solution to the SDE $dX_t = dt + 2 \sqrt{X_t} \, dW_t$

Question

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

Answer 1

Hints:

1. Suppose that $(X_t)_{t \geq 0}$ is a solution to the SDE. Using Itô's formula show that $Y_t := \sqrt{X_t}$ satisfies $$dY_t = dW_t.$$ (See the second part of this answer to get a better understanding how to choose a suitable transform.)
2. Conclude from step 1 that $X_t := (\sqrt{x}+W_t)^2$ is a candidate for the solution.
3. Use Itô's formula to show that $(X_t)_{t \geq 0}$ from step 2 is indeed a solution to the given SDE.