Simultaneous exceedance of a bivariate Ornstein–Uhlenbeck process

The stationary companion of the quadrant animation. Two Ornstein–Uhlenbeck processes with covariance matrix R(t) = e^−|t|Σ, where Σ has unit diagonal and off-diagonal entries ρ, run over [0, T]. The rare event is a simultaneous exceedance of u and b₂u. The quadratic programme min x^⊤Σ⁻¹x over x ≥ (1, b₂) identifies three cases. For ρ < b₂, both coordinates are essential and high-level entry concentrates near the corner. For ρ > b₂, only the first is essential, with second-coordinate slack (ρ − b₂)u + O_P(1). At ρ = b₂, that slack is of order one and the second constraint still affects the leading constant. Conditional on an exceedance, the first entry time becomes uniform on [0,T] as u → ∞; it is not exactly uniform at finite levels. The high-level exceedance probability is asymptotically T ℋ u² P(X(0) > u b) with the appropriate multidimensional Pickands constant ℋ. From Dębicki, Hashorva & Wang, Extremes of vector-valued Gaussian processes, Theorem 2.1. The uniform-time limit follows by applying the theorem to each initial subinterval.

the plane (X₁(t), X₂(t)); first detected grid exceedances marked; grid segments with both endpoints inside in orange; each path ends at T or about 0.8 time units after detection

the same paths against time — X₁ (dark), X₂ (blue) — with thresholds; below, the histogram of the slack X₂(τ) − b₂u at detection and its leading order-u term

Cross-correlation ρ = 0.30 Levels: b₁ = 1 (fixed), b₂ = 0.50 Level u = 2.5

Exact stationary OU samples on a grid (AR(1) coefficient e^−dt, dt = 0.01, T = 6). A stationary start already above both levels counts as time-0 entry. Crossings between samples are missed, and detected entries can overshoot. The continuous-time theory describes the high-level limit; at these moderate levels the dashed slack line is a scale guide, not an exact mean. Search stops after 24 hits or 60 000 attempts.