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  "slug": "small-hurst-fbm-maximum",
  "title": "The small-Hurst limit of the expected maximum of fractional Brownian motion",
  "shortTitle": "How high does a rough random path climb? The small-Hurst limit of the fBm maximum",
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  "oneLine": "As fractional Brownian motion becomes very rough (Hurst parameter H → 0), √H times its expected maximum on [0, 1] tends to √2, settling a 2022 conjecture; a uniform grid of n points sees half of that maximum once H log n reaches ½.",
  "abstract": "Fractional Brownian motion B_H is the standard model of a rough random path, and its Hurst parameter H sets the roughness: small H means a very rough path, as in rough-volatility models where H is around 0.1. Its expected maximum M_H = E sup over [0, 1] of B_H grows like H^(−1/2) as H → 0, and Bisewski, Dębicki and Rolski (2022) conjectured that √H·M_H has a finite positive limit; the known bounds placed it between 1.128 and 1.695. This candidate proves that the limit is √2, with convergence of the rescaled maximum in every L^p and error orders on both sides. More generally, a Gaussian field whose canonical metric is asymptotically d^H on a compact Ahlfors s-regular space has limit √(2s), and every such space carries such fields; for compact Euclidean sets the upper box dimension gives an upper bound. The all-time maximum with linear drift has the same constant. For a uniform grid with n intervals and H log n → c, the sampled maximum times √H tends to an explicit F(c), equal to √c up to c = ½ and then rising to √2, so a grid captures half the continuum mean at c = ½ and all of it only when H log n → ∞; quasi-uniform nets in regular spaces obey √s·F(c/s). The paper also proves M_H ≤ √(3 log 2 / H) for every 0 < H ≤ 1, with a consequent bound for Pickands' constant, and gives numerically evaluated two-sided bounds at fixed H and a Monte Carlo check of the grid law; these show that the first-order formulas converge slowly. Hierarchical Gaussian maxima (GREM/CREM), Gaussian comparison and concentration are classical; the vanishing-exponent limits, the dimension law and the grid crossover are the proposed contribution. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-03",
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  "keywords": [
    "fractional Brownian motion",
    "expected supremum",
    "Hurst parameter",
    "Gaussian processes",
    "Ahlfors regular spaces",
    "generalised random energy model",
    "rough volatility",
    "discretisation error"
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    "Small-Hurst limit (Theorem 1, proof): lim_{H→0} √H·E sup_{0≤t≤1} B_H(t) = √2, with L^p convergence of √H·sup B_H for every p, and √(2/H) − O(√log(1/H)) ≤ M_H ≤ √(2/H) + O(H^{3/2}); this settles the conjecture of Bisewski, Dębicki and Rolski (2022).",
    "Explicit bounds (Corollary 3 and Lemma 4, proof; Table 1, numerical): M_H ≤ √(3 log 2 / H) for every 0 < H ≤ 1, Pickands' constant H_H ≤ (29.86 H)^{1/(2H)} for 0 < H ≤ ½, and two-sided tree comparisons evaluated numerically (at H = 0.1, M_H lies in about [2.43, 3.50], against √(2/H) = 4.47).",
    "Dimension law (Theorem 5, Corollary 6, Proposition 7, proof): fields with canonical metric asymptotically d^H on a compact Ahlfors s-regular space have limit √(2s) (√(2d) for a cube in R^d); every such space carries such fields; for compact K ⊂ R^d the lim sup is at most √(2·upper box dimension), and Hausdorff dimension does not govern the limit.",
    "Drift (Theorem 8, proof): M_H − 1 ≤ E sup_{t≥0}(B_H(t) − t) ≤ M_H + 4.083 for H ≤ ½, so the all-time maximum with any linear drift has the same constant √2.",
    "Grid law (Theorems 11 and 13, proof; Table 2, Monte Carlo): with H log n → c, √H·E max over the grid {i/n} tends to F(c) = √c for c ≤ ½ and √2(1 − e^{1/2−c}/2) for c ≥ ½, so a grid captures half the continuum mean at c = ½ and 90% at c ≈ 2.11; quasi-uniform nets in Ahlfors s-regular spaces give √s·F(c/s). At H = 0.05–0.2 simulated grid means lie 17–42% below the first-order values."
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