If $X$ and $Y$ are two correlated Gaussian Random variables with parameters $(\mu_X, \sigma_X)$ and $(\mu_Y, \sigma_Y)$. If $Z=\exp(X - Y)$ then what is the distribution of $Z$.
Math world seems to suggest $X-Y$ should have a Normal Difference Distribution, based on this would that mean $Z$ has a log normal difference distribution ? I am unsure if there is any distribution named that. Also would the mean and standard deviation of $Z$ be simply $\exp(\mu(X - Y))$ and $\exp(\sigma(X - Y))$ where $\mu$ and $\sigma$ imply the mean and standard deviations one gets from Normal Difference distribution of $X-Y$.
Any help would be much appreciated.