$dS_t=\mu S_t dt +\sigma S_t^{\alpha/2} d W_t$ and $Y_t=S^{2-\alpha}$, can one simulate exact paths for Y_t?

Question

The task states that $dS_t=\mu S_t dt +\sigma S_t^{\alpha/2} d W_t$ and the question is if one can generate (simulate) exact paths for $S_t$ by taking the transformation s.t. $Y_t=S_t^{2-\alpha}$. I apply Ito's rule and then the dynamics of $Y_t$ are: $dY_t=(2-\alpha)*[\mu S_t^{2-\alpha} dt+\sigma S_t^{1-\alpha/2 }dW_t] +0.5 (2-\alpha)(1-\alpha)\sigma^2 dt$ and that is where I am stuck, because I know that for exact paths, dY_t needs to be independent of Y_t/S_t. But I am not sure where and if I made a mistake. What am I missing?

Answer 1

The question seems to be if that SDE has a known closed form solution. The answer is no, unless $\alpha=2$ (where it becomes a GBM).

The formula that Ito's rule yields you can rewrite as $$ dY_t=(2-\alpha)\Big[\mu Y_t\,dt+\sigma \sqrt{Y_t}\,dW_t\Big]+\frac{1}{2}(2-\alpha)(1-\alpha)\,\sigma^2\,dt\,. $$ This is formally equivalent to the interest rate model of Cox, Ingersoll and Ross: $$ dr=a(\theta-r)\,dt+\sigma\sqrt{r_t}\,dW_t $$ which has to date no known closed form solution.