Explicit estimation of the barrier-hitting probability in the Rough Bergomi model

Derive an explicit non-asymptotic bound for the barrier-hitting probability P(sup_{t ∈ [0,T]} S_t ≥ B) in the Rough Bergomi model, where S_t = exp(X_t) and X_t = x − (1/2)∫_0^t σ_s^2 ds + ∫_0^t σ_s dZ_s with Z_t = ρW_t + √(1−ρ^2)B_t and σ_t^2 = σ_0^2 exp(ν W^H_t − (ν^2 t^{2H})/2), W^H_t = √(2H)∫_0^t (t−s)^{H−1/2} dW_s. The bound should be valid without imposing uniform boundedness on σ, and should be expressed explicitly in terms of T, B, x, ρ, ν, H, and σ_0 to quantify the short-time decay rate.

Background

The main results of the paper on short-time decay of up-and-in barrier option prices rely on the assumption that the volatility process satisfies uniform bounds α ≤ σ_t ≤ β. This boundedness is crucial for deriving explicit tail and density bounds for the supremum of the log-price and, therefore, for the barrier-hitting probability.

In the Rough Bergomi model, the volatility is lognormal and does not satisfy such uniform boundedness. The authors therefore use a truncation/approximation argument to establish that the barrier-hitting probability still decays faster than any polynomial in T, but they do not obtain an explicit estimate of the hitting probability itself in the untruncated model. This leaves open the task of providing direct, explicit bounds for P(sup_{t ∈ [0,T]} S_t ≥ B) under Rough Bergomi dynamics without bounded volatility assumptions.