Simultaneous ruin probability for two-dimensional brownian risk model

Abstract

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.

Pandemic-type failures in multivariate Brownian risk models

“…The next result extends the findings of Dȩbicki et al (2020) to the case d > 2. For notational simplicity we consider the case I has d elements and thus avoid indexing by I. Recall that in our model W (t) = B(t) where B(t) has independent standard Brownian motion components and is a d × d non-singular real-valued matrix.…”

On the cumulative Parisian ruin of multi-dimensional Brownian motion risk models

“…for the inverse matrix of M II whenever it exists. As we will see, the solution to the quadratic programming problem involved in (6) is the key to our discussions.…”

“…, B d (t)) ⊤ , t ≥ 0 is a standard d-dimensional Brownian motion with independent coordinates. Multi-dimensional Brownian motion risk models have drawn a lot of attention due to its tractability; see, e.g., [6,10] and references therein.…”

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

“…A need to consider the joint survival function for (Q 1 , Q 2 ) appeared also in Lieshout and Mandjes [16] who considered two parallel queues sharing the same Brownian input (which is the case of ρ = 1) and also a Brownian tandem queue. We refer to [17] for further discussions on Gaussian-related queueing models and to [3,6] for the analysis of a related simultaneous ruin problem for the correlated Brownian motion model.…”