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

Abstract

AbstractFor $$\{\varvec{B}_{H}(t)= (B_{H,1}(t) ,\ldots ,B_{H,d}(t))^{{\top }},t\ge 0\}$$ { B H ( t ) = ( B H , 1 ( t ) , … , B H , d ( t ) ) ⊤ , t ≥ 0 } , where $$\{B_{H,i}(t),t\ge 0\}, 1\le i\le d$$ { B H , i ( t ) , t ≥ 0 } , 1 ≤ i ≤ d are mutually independent fractional Brownian motions, we obtain the exact asymptotics of $$\mathbb P (\exists t\ge 0: A \varvec{B}_{H}(t) - \varvec{\mu }t >\varvec{\nu }u), \ \ \ \ u\rightarrow \infty ,$$ P ( ∃ t ≥ 0 : A B H ( t ) - μ t > ν u ) , u → ∞ , where A is a non-singular $$d\times d$$ d × d matrix and $$\varvec{\mu }=(\mu _1,\ldots , \mu _d)^{{\top }}\in \mathbb {R}^d$$ μ = ( μ 1 , … , μ d ) ⊤ ∈ R d , $$\varvec{\nu }=(\nu _1, \ldots , \nu _d)^{{\top }} \in \mathbb {R}^d$$ ν = ( ν 1 , … , ν d ) ⊤ ∈ R d are such that there exists some $$1\le i\le d$$ 1 ≤ i ≤ d such that $$\mu _i>0, \nu _i>0.$$ μ i > 0 , ν i > 0 .