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

Abstract

This paper derives an exact asymptotic expression for Pxu{∃t≥0X(t)−μt∈U},asu→∞,where X(t)=(X1(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=∏i=1d[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.