On the upper bound in Varadhan's Lemma

In this paper, we generalize the upper bound in Varadhan's Lemma. The standard formulation of Varadhan's Lemma contains two important elements, namely an upper semicontinuous integrand and a rate function with compact sublevel sets. However, motivated by results from queueing theory, we do not assume that rate functions have compact sublevel sets. Moreover, we drop the assumption that the integrand is upper semicontinuous and replace it by a weaker condition. We prove that the upper bound in Varadhan's Lemma still holds under these weaker conditions. Additionally, we show that only measurability of the integrand is required when the rate function is continuous.Comment: 5 page

An optimization approach to adaptive multi-dimensional capital management

Firms should keep capital to offer sufficient protection against the risks they are facing. In the insurance context methods have been developed to determine the minimum capital level required, but less so in the context of firms with multiple business lines including allocation. The individual capital reserve of each line can be represented by means of classical models, such as the conventional Cram\'{e}r-Lundberg model, but the challenge lies in soundly modelling the correlations between the business lines. We propose a simple yet versatile approach that allows for dependence by introducing a common environmental factor. We present a novel Bayesian approach to calibrate the latent environmental state distribution based on observations concerning the claim processes. The calibration approach is adjusted for an environmental factor that changes over time. The convergence of the calibration procedure towards the true environmental state is deduced. We then point out how to determine the optimal initial capital of the different business lines under specific constraints on the ruin probability of subsets of business lines. Upon combining the above findings, we have developed an easy-to-implement approach to capital risk management in a multi-dimensional insurance risk model

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

We derive the exact asymptotics of ${\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \ \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} },\ \ u\to \infty ,$ where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena