Showing a process satisfies an SDE

Question

The example of Ito and Watanabe in the following notes

http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf

is an SDE without unique solutions.

$$dX_t = 3X_t^{1/3} dt + 3X_t^{2/3} dW_t$$

I am having trouble showing that $X_t = W_t^3$ satisfies this - I must be missing something basic. Can anyone help? Thanks!

Answer 1

Itô's formula states that

$$f(W_t)-f(0) = \int_0^t f'(W_s) \, dW_s + \frac{1}{2} \int_0^t f''(W_s) \,ds$$

for any $f \in C^2$. Applying this for $f(x) := x^3$ yields

$$X_t = 3 \int_0^t W_s^2 \, dW_s + 3 \int_0^t W_s \,ds.$$

Now the claim follows from the fact that $W_s^2 = X_s^{2/3}$ and $W_s = X_s^{1/3}$.