Ruin in every block, ruin in every window

The loss process X(t) = B(t) − ct of a Brownian risk model is watched over a horizon T = mh cut into m reporting blocks of length h. Block ruin asks for an exceedance of the level u in every block; sliding-window ruin asks for one in every window of length h, wherever it is placed — equivalently, no recovery interval of length h. Both probabilities are bounded by first-block ruin and hence by ordinary ruin on [0, T]; they are not products of one-block probabilities. As u → ∞ with h, c, m fixed, conditional on ruin in the first block, the first hit of u concentrates just before h; the strong Markov property there turns everything that follows into a finite-level event, so p^bl ∼ 2 G_m Ψ((u+ch)/√h) and p^sl ∼ 2 G^sl_m Ψ((u+ch)/√h), where G_m is the probability that a fresh drifted Brownian motion is positive somewhere in each of the blocks [kh,(k+1)h], k = 1,…,m−2, and G^sl_m the Parisian-type survival probability that no length-h interval within [0,(m−1)h] is entirely non-positive. The animation rains down Brownian paths until one achieves the event, then replays it. From Dębicki, Hashorva & Michna, Finite-horizon block and sliding-window ruin for Brownian risk models.

the loss X(t) = B(t) − ct over the m blocks; the level u dashed; failed attempts in grey; the first exceedance in each block marked, and the first hitting time τ_u — which crowds against h as u grows

frequencies for the piecewise-linear simulated paths, with Wilson uncertainty bars, against the continuous-path asymptotics 2GΨ((u+ch)/√h); ψ(u,h) is the exact continuous first-block probability

grid-estimated first hitting times τ_u of all paths ruined in block 1, whether or not later blocks are ruined; the curve is the continuous-time conditional density (inverse Gaussian restricted to [0,h]); the lemma says τ_u | τ_u < h → h

Level u = 1.5 Premium rate c = 1.0 Block length h = 1.0 · blocks m = 4 Replay speed ×1.0

Brownian increments are sampled at step h/200 and joined by straight lines. Block maxima and recovery intervals are evaluated on this polygonal path. The constants G_m, G^sl_m and J_m = P{B(jh) − cjh > 0, 1 ≤ j ≤ m−2} use 4 000 such paths; for m = 2 their exact values are 1. The estimates can be noisy or zero. The theoretical ordering is J_m ≤ G^sl_m ≤ G_m ≤ 1. Interpolation misses Brownian fluctuations between samples; the sliding-event error has no asserted direction. A fixed mesh does not validate a high-level limit, and 2GΨ is an asymptotic, not a finite-level identity. Wilson bars are the usual fixed-sample approximation for sampling uncertainty; they do not account for numerical error or provide a sequential confidence guarantee. Brute-force search becomes slow when first-block ruin is rare.