Extremal behaviour of hitting a cone by correlated Brownian motion with drift

This paper derives an exact asymptotic expression for Pxu {∃t≥0X(t) − μt ∈ U}, as u → ∞, where X(t) = (X₁1(t), . . . , Xd (t))⊤, t ≥ 0 is a correlated d-dimensional Brownian motion starting at the point xu = −αu with α ∈ Rd , μ ∈ Rd and U = Πd i=1[0,∞). The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to U, which depends on the dimension reduction phenomena