Asymptotic expansions for the moments of the Gaussian random walk with two barriers

In this study, the semi-Markovian random walk (X(t)) with a normal distribution of summands and two barriers in the levels 0 and [beta]>0 is considered. Moreover, under some weak assumptions the ergodicity of the process is discussed and the characteristic function of the ergodic distribution of the process X(t) is expressed by means of appropriating one of a boundary functional SN. Using this relation, the exact formulas for the first four moments of ergodic distribution are obtained and the asymptotic expansions are derived with three terms for the one's, as [beta]-->[infinity]. Finally, using the Monte Carlo experiments, the degree of accuracy of obtained approximate formulas to exact one's have been tested.