2020/04/29 by Dȩbicki, Krzysztof, Hashorva, Enkelejd, Krystecki, Konrad
#60G15 (Primary) 60G70 (Secondary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2004.14015
Let (W1(s), W2(t)), s,t≥ 0 be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation ρ∈ (-1,1) and define the joint survival probability of both supremum functionals πρ(c1,c2; u, v) by πρ(c1,c2; u, v)=ℙ(sups ∈ [0,1] (W1(s)-c1s)gt;u,supt ∈ [0,1] (W2(t)-c2t)gt;v) , where c1,c2 ∈ ℝ and u,v are given positive constants. Approximation of πρ(c1,c2; u, v) is of interest for the analysis of ruin probability in bivariate Brownian risk model as well as in the study of bivariate test statistics. In this contribution we derive tight bounds for πρ(c1,c2; u, v) in the case ρ∈ (0,1) and obtain precise approximations by letting u→ ∞ and taking v= au for some fixed positive constant a and ρ∈ (-1,1).