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

Abstract

We analyze the classical Brownian risk models discussing the approximation of ruin probabilities (classical, γ-reflected, Parisian and cumulative Parisian) for the case that ruin can occur only on specific discrete grids. A practical and natural grid of points is for instance G(1) = {0, 1, 2, . . .}, which allows us to study the probability of the ruin on the first day, second day, and so one. For such a discrete setting, there are no explicit formulas for the ruin probabilities mentioned above. In this contribution we derive accurate approximations of ruin probabilities for uniform grids by letting the initial capital to grow to infinity.

Simultaneous Ruin Probability for Two-Dimensional Fractional Brownian Motion Risk Process over Discrete Grid

“…The asymptotics in Theorem 3.1 for H = 1/2 reminds the results in [19]; the only difference is in Pickands-type constants. For case H < 1/2 due to short-range dependence of fBm asymtpotics of π Z,H (u) as u → ∞ coincides with the asymptotics of the classical discrete ruin probability, see [14,20].…”

“…Then the claim follows from the proof of Theorem 1.1 in [7]. The index of summation in (19) in [7] in our case will be 1 j N u , and thus the factor 1/2 appears before the final asymptotics. The claim can also be established by Theorem 1(ii) in [10].…”

Simultaneous ruin probability for two-dimensional fractional Brownian motion risk process over discrete grid, with supplements

“…Proof of Theorem 2.4. When H = 1 2 the assertion of the theorem follows from the results in [6] and [8].…”

“…By Borell-TIS inequality (Lemma 5.3 in [7]), (see also proof of Theorem 1.1 in [6] and proof of (26)) we…”

Parisian ruin with power-asymmetric variance near the optimal point with application to many-inputs proportional reinsurance

“…Another interesting univariate case is of Parisian ruin over discrete sets. In [12], the authors have proved that for the Brownian motion and equidistant grid the asymptotics differs from the continuous one by some constant factor.…”