I'm trying to solve the following nonlinear stochastic differential equation
$$ dx = 3a(x^{1/3}-x)dt + 3\sqrt{a}x^{2/3}dW $$
According to my TA, I'm supposed to transform variables to $y=x^{1/3}$, which I do by Ito's formula:
$$ dy = \frac{1}{3}x^{-2/3} dx - \frac{2}{9}x^{-5/3}dx^2 $$
Inserting the expression for $y$ and $dx$ I get (disregarding factors larger than $dt$):
$$ dy = -a(y^{-1} + y)dt + \sqrt{a}dW $$
I'm supposed to solve for $y$ and then transform back to get the solution for $x$. But, solving the equation for $y$ above doesn't seem straightforward to me at all.
Do you have any input?
The variable transformation is right. But, there is an error in your computation of $dy$ using the Ito's formula. It should read
$$ dy = \frac{1}{3}x^{-2/3} dx - \frac{1}{9}x^{-5/3}dx^2 $$
As a result, you should obtain the following standard OU process
$$ dy = -aydt + \sqrt{a}dW $$
To solve it, rewrite
$$d(e^{at}y)=e^{at}\sqrt{a}dW$$
with its solution given by
$$y_t=y_0e^{-at} + \sqrt{a}\int_0^te^{-a(t-s)}dW_s$$
The solution for $x$ is then
$$x_t = \left( x_0^{1/3}e^{-at} + \sqrt{a}\int_0^te^{-a(t-s)}dW_s \right)^3 $$