On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function

TL;DR

This paper derives the inverse Laplace transforms of products of parabolic cylinder functions related to the Ornstein-Uhlenbeck process, using probabilistic methods and recurrence relations, enabling broader applications in stochastic process analysis.

Contribution

It provides the first documented inverse transforms of specific products of parabolic cylinder functions linked to the Ornstein-Uhlenbeck process, expanding the analytical tools available for such functions.

Findings

Inverse transforms of D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y) are derived.

Recurrence relations facilitate inverse transforms for other combinations of parabolic cylinder functions.

Results connect Laplace transforms with transition densities of stochastic processes.

Abstract

The Laplace transforms of the transition probability density and distribution functions for the Ornstein-Uhlenbeck process contain the product of two parabolic cylinder functions, namely D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y), respectively. The inverse transforms of these products have as yet not been documented. However, the transition density and distribution functions can be obtained by alternatively applying Doob's transform to the Kolmogorov equation and casting the problem in terms of Brownian motion. Linking the resulting transition density and distribution functions to their Laplace transforms then specifies the inverse transforms to the aforementioned products of parabolic cylinder functions. These two results, the recurrence relation of the parabolic cylinder function and the properties of the Laplace transform then enable the calculation of inverse transforms also for countless other combinations in the orders of the parabolic cylinder functions such as D_{v}(x)D_{v-2}(y), D_{v+1}(x)D_{v-1}(y) and D_{v}(x)D_{v-3}(y).