If $S(t)$ is the stock price that satisfies BSM model in SDE form how can I derive an SDE for $S^n (t)$ for some positive integer n

Question

If $S(t)$ is the stock price that satisfies BSM model in SDE form where

$dS(t) = \mu S(t) dt + \sigma S(t) d W(t)$ where $\mu >0$ and $\sigma>0$ are two constants.

how can I derive an SDE for $S^n (t)$ for some positive integer n. Can I say that $S^n (t)$ is a geometric brownian motion?

Would someone mind helping me with this problem, I am studying for a test and any help would be greatly appreciated.

Answer 1

Let $S(t)$ be governed by the SDE

$$dS(t)=\mu S(t)dt+\sigma S(t)dW_t$$

Let $f(S)=S^n$.

Heuristically, we can write

$$\begin{align} d(S^n)&=\frac{\partial f(S)}{\partial t}\,(dt)+\frac{\partial f(S)}{\partial S}\,(dS)+\frac12\frac{\partial ^2f(S)}{\partial S^2}(dS)^2\\\\ &=\frac{\partial S^n}{\partial t}\,(dt)+\frac{\partial S^n}{\partial S}\,(dS)+\frac12\frac{\partial ^2 S^n}{\partial S^2}(dS)^2\\\\ &=(0)\,dt+nS^{n-1}\,(dS)+\frac12 n(n-1)S^{n-2}(dS)^2\\\\ &=\left(n\mu+\frac12 n(n-1)\sigma^2\right)S^n\,dt+n\sigma S^n dW_t \tag 1 \end{align}$$

Therefore, $S^n$ does follow a GBM process with drift $n\mu+\frac12 n(n-1)\sigma^2$ and diffusion coefficient $n \sigma$