  The goal of this paper is to derive a formula for the finite dimensional
joint characteristic function (the Fourier transform of the finite dimensional
distribution) of the coupled process ${(W_{t},L_{t}^{A}):t\in \lbrack
0,\infty)}$, where $\{W_{t}:t\in \lbrack 0,\infty)}$ is a $d$-dimensional
Brownian motion and $\{L_{t}^{A}:t\in \lbrack 0,\infty)}$ is the generalized
$d$-dimensional L$\acute{e}$vy's stochastic area process associated to a
$d\times d$ matrix $A.$ Here $A$ need not be skew-symmetric, and in our
computation we allow $A$ to vary. The problem finally reduces to the solution
of a recursive system of symmetric matrix Riccati equations and a system of
independent first order linear matrix ODEs. As an example, the two dimensional
L\'{e}vy's stochastic area process is studied in detail.
