A Diffusion Equation for the Density of the Ratio of Gaussian Variables and the Numerical Inversion of Laplace Transform

Introduction. The density of the ratio of two random variables with joint bivariate Gaussian density has been derived by several authors and it is important in many applications (see e.g. [8, 13, 14, 15]). In the sequel it is proved that, when the two variables have the same variance, this density satisfies a parabolic partial differential equation whose coefficients depend on both the independent variables. The proof is based on standard properties of the confluent hypergeometric functions of the first kind. A motivation for deriving such a PDE is provided by the problem of the numerical inversion of Laplace transform from noisy discrete data [2, 4]. This is a classical ill-posed problem. Insights for its stable solution can be obtained from knowledge of the marginal densities of the damping factors of a multiexponential model which represents a discretization of the Laplace transform. This problem can be restated in terms of the condensed density of the generalized eigenvalues of a matrix pencil built from the observations. In a recent paper [5] an adaptive kernel density estimator based on linear diffusion processes has been proposed which has several advantages over the existing methods. In the sequel a kernel density estimator in the class considered in [5], based on the proposed diffusion equation, for estimating the condensed density mentioned above is proposed. A Montecarlo simulation allows to appreciate its merits with respect to a Gaussian kernel estimator and its effectiveness for the numerical inversion of the Laplace transform. The paper is organized as follows. In the first section the density of the ratio of two random variables with joint bivariate Gaussian density is shortly