Expectation and Variance of Modified Geometric Brownian Motion

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 $$\sigma^2 t$$?

• Have you actually tried deriving the closed form solution for this? – Makina Dec 20 '18 at 23:52
• Yes, but my knowledge of solving SDEs is limited. I tried the same ansatz for GBM, that is Y = ln X, but it did not work. It seems quite difficult to solve SDEs in general, unless you can guess the form of the solution. – QuantopianDreams Dec 21 '18 at 18:21
• Hi @QuantopianDreams, look up linear SDEs. – AddSup Dec 22 '18 at 6:48