Staying high everywhere: the infimum of fractional Brownian motion

How does a fractional Brownian motion manage to stay above a high level u on a whole set 𝒦 = [a₁,b₁] ∪ [a₂,b₂]? The exponential rate is set by an optimal measure μ_* minimising Var ∫X dμ, and its potential m(t) = Cov(X(t), X_μ*)/Var X_μ* ≥ 1; the contact set {m = 1} decides the prefactor. For H > ½ the measure is the point mass at a₁, the contact set is that single point: on the scale of u, the typical high path rises away from that endpoint — dominated contact, with P ~ P(B_H(a₁) > u). For Brownian motion the potential is ≡ 1 on all of 𝒦: full contact, the leading path divided by u is flat there, while order-one upward fluctuations remain possible. A Laplace-transform reduction gives an extra factor asymptotic to κ a₁/u relative to the one-point tail. From Dębicki, Hashorva & Novikov, Gumbel MDA & tail behaviour of infima of Gaussian processes.

paths of B_H (dim), the first 40 paths with inf_𝒦 B_H > u (dark); 𝒦 shaded. Lines join sampled values; Brownian crossing tests include the bridges.

the potential m(t) − 1 on 𝒦 (top) and the ratio P(inf > u) / asymptotic against u (bottom)

Monte Carlo P(inf_𝒦 > u)

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Asymptotic

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Contact set

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Hurst index H = 0.50 Level u = 2.0

𝒦 = [1, 2] ∪ [2.5, 3.5] (or [1, 2] alone). fBm by Cholesky on a grid including every interval endpoint; 60 000 paths stored so the level slider re-reads them. At H = ½, independent Brownian-bridge minima give exact continuous-interval crossing tests. At H > ½, the grid infimum overestimates the continuous one, so the estimate has upward discretisation bias; the bias need not be small at high levels. The constant κ_A,G,C is computed numerically from the paper's double integral.