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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)?

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$$E(X+Y) = E(X) + E(Y)$$ and $$\sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2 + 2 \rho \sigma_X \sigma_Y$$

where $\rho$ is the correlation coefficient.

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