With {ξ i } i≥0 being a centered stationary Gaussian sequence with non negative correlation function ρ(i) := E [ξ 0 ξ i ] and {σ(i)} i≥1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ℓ i=1 σ(i)ξ i , ℓ ≥ 1. For summable correlations ρ, we show that the persistence exponent is universal. On the contrary, for non summable ρ, even for polynomial weight functions σ(i) ∼ i p the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ(H, p) and study its properties as a function of (p, H). During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non negative correlations -e.g. a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero -that might be of independent interest.
We study the asymptotics of the ruin probability in the Cramér-Lundberg model with a modified notion of ruin. The modification is as follows. If the portfolio becomes negative, the asset is not immediately declared ruined but may survive due to certain mechanisms. Under a rather general assumption on the mechanism -satisfied by most such modified models from the literature -we study the relation of the asymptotics of the modified ruin probability to the classical ruin probability. This is done under the Cramér condition as well as for subexponential integrated claim sizes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.