Probability of entering an orthant by correlated fractional Brownian motion with drift: exact asymptotics (Q6635939)

Let \(\tilde B_H=\{(B_{H,1}(t),\dots,B_{H,d}(t)),t\geq 0\}\) be a multi-dimensional fractional Brownian motion where \(B_{H,i}, 1\leq i \leq d\) are independent one-dimensional fractional Brownian motions with common Hurst index \(H.\) The authors study the exact asymptotic behaviour of the probability that there exists \(t\geq 0\) such that \(A \tilde B_H(t)-\tilde \mu t >\tilde v u\), where \(A\) is a non-singular \(d\times d\) matrix \(\tilde \mu= (\mu_1,\dots,\mu_d)^\prime \in \mathbb{R}^d\) and \(\tilde v= (v_1,\dots,v_d)^\prime \in \mathbb{R}^d\) are such that there exists some \(1\leq i \leq d\) such that \(\mu_i>0\) and \(v_i>0.\) The study focuses on the exact asymptotic behaviour of the probability that a drifted correlated fractional Brownian motion \(\tilde X\) enters the orthant \(O_u=\{(x_1,\dots,x_d)^\prime:x_i>v_iu,i=1,\dots,d\}\) over an infinite-time horizon. They consider the case when the probability \( P(u)\) described above is a rare event, that is, \(P(u) \rightarrow 0\) as \(u\rightarrow \infty\) assuming that there exists some \(1\leq i \leq d\) such that \(\mu_i>0\) and \(v_i>0\). The problem discussed here has applications in the subject of ruin theory. The result obtained here is an extension of the result proved in [\textit{J. Hüsler} and \textit{V. Piterbarg}, Stochastic Processes Appl. 83, No. 2, 257--271 (1999; Zbl 0997.60057)]. Since there are no Slepian-type inequalities that can be used in the multi-dimensional setting, the authors use the techniques developed in [\textit{K. Dȩbicki} et al., Stochastic Processes Appl. 128, No. 12, 4171--4206 (2018; Zbl 1417.60028)].