arXiv subjects

Dirk Veestraeten

Publications and source records attributed to Dirk Veestraeten.

5 recordsLinked to original sources

arXiv2019

A new integral equation for the first passage time density of the Ornstein-Uhlenbeck process

The Laplace transform of the first passage time density of the Ornstein--Uhlenbeck process for a constant threshold contains a ratio of two parabolic cylinder functions for which no analytical inversion formula is available. Recently derived inverse Laplace transforms for the product of two parabolic cylinder functions together with the convolution theorem of the Laplace transform then allow to derive a new Volterra integral equation for this first passage time density. The kernel of this integral equation contains a parabolic cylinder function and the Fortet renewal equation for the Ornstein-Uhlenbeck process emerges as a special case, namely when the order q of the parabolic cylinder function is set at 0. The integral equation is shown to hold both for constant as well as time dependent thresholds. Moreover, the kernel of the integral equation is regular for q<=-1.

math.PR↗

arXiv2019

Some Laplace transforms and integral representations for parabolic cylinder functions and error functions

This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently specialized for products of the error function and its complement thereby yielding new integral representations for products of the latter two functions. The transforms that are derived in this paper also allow to correct two inverse Laplace transforms that are widely reported in the literature and subsequently uses one of the corrected expressions to obtain two new definite integrals for the generalized hypergeometric function.

math.CA↗

arXiv2015

Integral representations for products of two parabolic cylinder functions with different arguments and orders

This paper derives new integral representations for products of two parabolic cylinder functions. In particular, expressions are obtained for D_{nu}(x)D_{mu}(y), with x>0 and y>0, that allow for different orders and arguments in the two parabolic cylinder functions. Also, two integral representations are obtained for D_{nu}(-x)D_{mu}(y) by employing the connection between the parabolic cylinder function and the Kummer confluent hypergeometric function. The integral representations are specialized for products of two complementary error functions and of two modified Bessel functions of the second kind of order 1/4, as well as for the product of a parabolic cylinder function and a modified Bessel function of the first kind of order 1/4.

math.CA↗

arXiv2015

Some integral representations and limits for (products of) the parabolic cylinder function

Veestraeten [1] recently derived inverse Laplace transforms for Laplace transforms that contain products of two parabolic cylinder functions by exploiting the link between the parabolic cylinder function and the transition density and distribution functions of the Ornstein-Uhlenbeck process. This paper first uses these results to derive new integral representations for (products of two) parabolic cylinder functions. Second, as the Brownian motion process with drift is a limiting case of the Ornstein-Uhlenbeck process also limits can be calculated for the product of gamma functions and (products of) parabolic cylinder functions. The central results in both cases contain, in stylised form, D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y) such that the recurrence relation of the parabolic cylinder function straightforwardly allows to obtain integral representations and limits also for countless other combinations in the orders such as D_{v}(x)D_{v-3}(y) and D_{v+1}(x)D_{v}(y).

math-ph↗

arXiv2015

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

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).

math-ph↗