Approximation of Sojourn Times of Gaussian Processes

Abstract

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous-and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of [1, 2] for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.

Approximation of ruin probability and ruin time in discrete Brownian risk models

“…We approximate each summand in p 1 (S, u) uniformly. As in the proof of Theorem 1.1 we obtain and ω(j, S, x) is defined in (17). By Borell-TIS inequality (similarly the proof of (18) It follows with similar arguments as in [24] that as S → ∞…”

“…with I(·) denoting the indicator function. In view of [17] B η (k) is positive and finite. follows with similar arguments as in the proof of Theorem 1.1 that those can be approximated in the same way as (5).…”

“…and ω(i, S, x) is defined in (17). By Borell-TIS inequality for all |i|, |j| ≤ N u (proof is in the Appendix)…”

“…Section 3 discusses the ruin probability for the γ-reflected Brownian risk model, see also [9][10][11][12]. The approximation of Parisian ruin (see [13][14][15]) and cumulative Parisian ruin (see [16][17][18])…”

On Generalized Berman Constants

“…The recent contribution [3] discussed (1.1) and gave the approximations of the related sojourn time of discrete form for locally stationary Gaussian processes, and [4] investigated general Gaussian processes with strictly positive drift function. For more related discussions on ruin time and the extremal analysis of Gaussian processes and random fields in financial and insurance framework, we refer to [5][6][7][8][9][10][11][12][13][14].…”