Stochastic DEs.

Question

If X_t solves dX_t = μX_t dt + σ dW_t, find the stochastic differential equation solved by:

1. X_tⁿ, where n is a positive integer.
2. cos X_t.

Any help is appreciated.

Answer 1

Let $f(x)=x^{n}$ where $n>1$. Note that $f_{x}(x)=nx^{n-1}$ and $f_{xx}(x)=n(n-1)x^{n-2}$. By Ito's lemma, \begin{align*} d(X_{t}^{n}) & =f_{x}(X_{t})dX_{t}+\frac{1}{2}f_{xx}(X_{t}) dX_t^2\\ & =nX_{t}^{n-1}\left(\mu X_{t}dt+\sigma dW_{t}\right)+\frac{1}{2}n\left(n-1\right)X_{t}^{n-2}\sigma^{2}dt\\ & =\left(\mu nX_{t}^{n}+\frac{1}{2}\sigma^{2}n\left(n-1\right)X_{t}^{n-2}\right)dt+\left(\sigma nX_{t}^{n-1}\right)dW_{t}. \end{align*} This is an SDE satisfied by $(X^{n}_t)_{t\geq0}$.

Now, try the second question.