The most likely path of a large exponential integral

Let X be a fractional Brownian motion with Hurst index H on [0,1] and I = ∫₀¹ e^X(t) dt. How does I become larger than a big b? The answer is a variational problem in the reproducing kernel Hilbert space: the cheapest path h_b with ∫e^h_b ≥ b, whose norm ρ_b gives the Jensen–Gaussian lower bound P(I > b) ≥ Ψ(ρ_b). The variational path concentrates its exponential mass near the variance-maximising endpoint t = 1; below we compare a numerical candidate with simulated exceedances. The bound is asymptotically exact when H > ½; for Brownian motion the ratio tends to √2; for rougher paths the cited result gives logarithmic asymptotics, without an exact ratio limit. From Hashorva & Novikov, Tail and density asymptotics for Gaussian exponential integrals.

sample paths of X (dim), the paths with I > b (dark), the numerical RKHS candidate h_b (orange dashed); up to 40 exceedances selected in simulation order

grid approximation to P(I > b) / Ψ(ρ_b) against log b; lines at 1 and √2 are continuous-time limits

Level b

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ρ_b and Ψ(ρ_b)

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Monte Carlo P(I > b)

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Hurst index H = 0.75 Level: log b = 2.0

Grid of 60 points; fBm by Cholesky; 40 000 paths, stored so that the level slider re-reads them instantly. The numerical candidate solves the discrete KKT system R⁻¹h = λ Δt e^h, Δt Σe^h = b by Newton continuation; stationarity alone does not certify global minimality. The continuous-time results require mesh refinement as b grows. The ratio curve stops below 20 exceedances; even its endpoint is statistically noisy.