On some boundary crossing problems for Gaussian random walks

On the identification of Wiener-Hopf factors

“…The left and right hand sides of equation (8) can be continued analytically into the domains lsl < 1 and lsl > l, respectively. Therefore there exists some entire function ·tjJ with which they coincide in these domains.…”

“…The technique is used there for calculating a stationary state distribution. But in addition to their usage in calculating stationary distributions, the Wiener-Hopf factors have a key role in the analysis of hitting times and other functionals of trajectories of random walks (Asmussen [l, Chapter VII], Borovkov [4,Chapter 3], Lotov and Khodjibayev [9], Lotov [8]). …”

“…Recent results concerning the identification of the Wiener-Hopf factors of Gaussian characteristic functions are given in Lotov [8]. These results are quite involved, but explicit.…”

On the First Crossing of Two Boundaries by an Order Statistics Risk Process

“…The probability that a stochastic process stays between two boundaries is of significant importance in many areas among which: statistics, for constructing confidence intervals for distribution functions, (see Wald and Wolfovitz 1939;Steck 1971) or in sequential analysis; finance, for pricing double-barrier options, (see Borovkov and Novikov 2005); actuarial science, in modelling the surplus of an insurance company (see Teunen and Goovaerts 1994), and also in probability (see e.g., Potzelberger and Wang 2001;Lotov 1996;Buonocore et al 1990) and other areas.…”

Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

“…By making use of the results in Siegmund (1982), (1988), Chang (1992) extended (1.4) to a high-order asymptotic expansion, with the o(θ) replaced by C a θ 2 + O(θ 3 ), where C a is a constant depending on a, the distribution of overshoot, and the renewal function of the descending ladder random variables. Further refinements of (1.4) can be found in Lotov (1996), and Chang and Peres (1997) for Gaussian random walks, to which the coefficients are related to the celebrated Riemann zeta function. Using different techniques to those utilized by Chang (1992), Blanchet and Glynn (2006) proposed a method to compute the coefficients in the asymptotic expansion of the moments of the first ladder heights for non-Gaussian random walks (up to arbitrary order).…”