how to write the joint density fuction of two variables that obey lognormal distribution

Question

Suppose $U_t$ is a random variable subject to $\operatorname{Lognormal}(x_1, z_1^2)$ distribution. $V_t$ is a random variable subject to $\operatorname{Lognormal}(x_2,z_2^2)$ distribution. Suppose their correlation coefficient is $p$. How to compute their joint density function?

Answer 1

This is not possible on the basis of the data given. Knowing the distribution of each variable, and their covariance, one does not have nearly enough information to determine the joint distribution.

To see why, consider the discrete situation: two variables taking values $1,\dots,n$. Their joint distribution is described by a matrix of size $n\times n$: that's $n^2$ unknowns. The distribution of each variable gives you $n$ linear equations, that's $2n$ altogether. And the correlation adds one more (nonlinear). You can't expect to determine $n^2$ unknowns from $2n+1$ equations.