Approximation formulas for the moments of the boundary functional of a Gaussian random walk with positive drift by using Siegmund's formula

Abstract

In this study, a boundary functional () are mathematically constructed for a Gaussian random walk (GRW) with positive drift beta and first four moments of the functional are expressed in terms of ladder variables based on Dynkin Principle. Moreover, approximation formulas for first three moments of ladder height are proposed based on the formulas of Siegmund (1979) when beta down arrow 0. Finally, approximation formulas for the first four moments of the boundary functional are obtained by using Siegmund formulas and meta modeling, when beta is an element of[0.1, 3.6].