How to solve the SDE: $dX_t = \frac{1}{3}X_t^{\frac{1}{3}}dt+X_t^{\frac{2}{3}}dB_t$?

Question

I am wondering how we can solve the SDE:

$$ dX_t = \frac{1}{3}X_t^{\frac{1}{3}}dt+X_t^{\frac{2}{3}}dB_t $$

for $X_0 = x >0$?

My approach is to use the Ito's Space and Time variable formula, but with that, I am still left with the equation:

$$ 3X_t^{\frac{1}{3}}\frac{df}{dt} + \frac{dX_t}{dB_t} = X_t^{\frac{2}{3}} $$

Anyone have any ideas how to solve this SDE? Thanks!

Answer 1

Hint: Use the transform $$Z_t := f(X_t) := X_t^{1/3}$$ to transform the given SDE into a (very simple) linear SDE for $(Z_t)_{t \geq 0}$.

Remark: If you want to know how I came up with this transform, have a look at "Solution 2" in this answer.