Consider the following stochastic differential equation (SDE):
$dX_t = (\mu X_t - q) dt + \sigma X_t dW$
What is the mean and variance of $X_t$? Is there an analytical solution for $X_t$?
This is a slightly modified Geometric Brownian Motion (GBM) process, with an additional $q$ term in the drift. Do any of the results of GBM still hold? Specifically, can we say that because only the drift term is modified, the variance term is the same as the case of GBM? Recall in GBM,
\begin{align} d ln X_t &= (\mu - \frac{1}{2} \sigma^2) dt + \sigma dW \\ \implies ln (\frac{X_{t}}{X_0}) & \sim N((\mu - \frac{1}{2} \sigma^2)t, \sigma^2 t) \end{align}
If we let $X_t$ represent the stock price, we can say that log returns are distributed normally with variance $\sigma^2 t$. Can we say that the log returns of the modified GBM process also has variance $\sigma^2 t$?