I'm sitting with a special case of GBM where $B(t)$ is Brownian motion with drift $\mu$ and variance $\sigma^2$. I want to find the expected value of $A(t)$, where $A(t)$ is given by $$A(t)=exp(B(t))B(t)+C$$ and C is a constant. Has anyone worked with such a model before?
I'm thinking that $$E[A(t)]=E[B(t) \exp(\mu t)]+C=\mu t\exp(\mu t)+C$$ is the answer to the expected value (not 100% sure), but how do I determine the variance of $A(t)$?
Thanks in advance.