Finding Stochastic processes

Question

I have the following differential equation

dX$_t$ = (r$\mu$X$_t$ + $\frac{r(r-1)}{2}σ^2X_t$)dt + rσX$_t$dB$_t$,

X$_0$ = x, with x > 0.

Here, r>0. I am having trouble figuring out how to find the process X$_t$. I would really appreciate help!

Answer 1

$$ dX_t=\left(r\mu+\frac{1}{2}r(r-1)\sigma^2\right)X_tdt+r\sigma X_t dB_t $$ By application of Ito lemma, we have $$d(\ln X_t)=\frac{1}{X_t}dX_t+\frac{1}{2}\left(\frac{-1}{X_t^2}\right)d[X_t,X_t]$$ therefore $$d(\ln X_t)=\left(r\mu+\frac{1}{2}r(r-1)\sigma^2\right)dt+r\sigma dB_t-\frac{1}{2}r^2\sigma^2dt$$ in other words $$d(\ln X_t)=\left(r\mu-\frac{1}{2}r\sigma^2\right)dt+r\sigma dB_t$$ thus $$\ln X_t-\ln x_0=\left(r\mu-\frac{1}{2}r\sigma^2\right)t+r\sigma B_t$$ finally we have $$X_t=x_0\exp\left(\left[r\mu-\frac{1}{2}r\sigma^2\right]t+r\sigma B_t\right)$$