# What is the characteristic formula for the addition of two lognormal distributions?

If I have two lognormal processes ($X$ and $Y$) with mean and volatiilty for each and also correlation between the two, what is the characteristic formula of $X + Y$ (i.e. what is the new mean and new volatility of $X + Y$)?

$$E(X+Y) = E(X) + E(Y)$$ and $$\sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2 + 2 \rho \sigma_X \sigma_Y$$
where $\rho$ is the correlation coefficient.