The Joint Density of Two Functionals of a Brownian Motion

In time series containing an autoregressive unit root, almost all known statistics can be described in terms of two Wiener functionals. It is therefore crucial for us to know how these functionals are jointly distributed in terms of explicit formulae that can be manipulated analytically, that do not contain integrals and that can be computed quickly and exactly. This paper shows how such distributions are arrived at; thus solving the problem, as well as paving the way for similar derivations in the future for the case of different stochastic processes. Also, a formula is provided to simplify the calculation of the distribution of any statistic that can be fully characterized in terms of these functionals, thus remedying the specific problem of unit-root distributions which have drastically different formula for each statistic.