Stochastic differential equation for $Y(t)=\sqrt{X(t)}$

Question

Assume that $X(t)$ solves the stochastic differential equation $$dX(t)=\sigma(t)dW(t)+\mu(t)dt$$ with $\mu(x)=bx+c$ and $\sigma^2(x)=4x.$ Assume that $X(t)\ge 0$. Find the stochastic differential equation for the process $Y(t)=\sqrt{X(t)}$.

My idea: I use Ito formula and get $df(X(t))=\frac{1}{2\sqrt{X(t)}}dX(t)-\frac{1}{8X(t)\sqrt{X(t)}}\sigma^2(X(t))$dt.

My problem is how to rearrange $Y(t)$ and use $\mu(x)=bx+c$ .

Thank you for help.

Answer 1

Use $\mathrm dX=\sigma(X)\mathrm dW+\mu(X)\mathrm dt$, $\mathrm d\langle X\rangle=\sigma(X)^2\mathrm dt$, the explicit formulas for $\sigma$ and $\mu$, and replace every $\sqrt{X}$ by $Y$ in Itô's formula for $\mathrm dY$, to deduce after some simplifications that $$ \mathrm dY=\mathrm dW+\tfrac12(bY+(c-1)Y^{-1})\,\mathrm dt. $$