Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal distribution. How can I derivate that for the mean $m(t)$ and variance $v(t)$ yields: $$ m(t)=\ln(X(0))+\left(rT-\frac{1}{2}v(t)\right) $$ and $$ v(t)=(\sigma_{0}^2-\sigma_{1}^2)t+\sigma_{1}T $$ where $T$ is the endtime.
My first idea was to work out the integrals, but I'm stuck with that. Hopefully anyone can help!