L²-optimal MA-fBM approximation error · Type II · T = 1
γ endpoints scale per-point as
exp(±α√K / ((d − H or H)·A(H))), with α = 1.06418 and A(II)(H) = √(1/H + 1/(d − H)), where d = 3/2 for H < 1/2 (rough regime, µ-density) or d = 5/2 for H > 1/2 (smooth regime, ν-density / U-basis analysis). See proof below.
Error vs K
0.3
empirical: \(\mathcal{E}^{(II)}(\omega^*)\)
predicted rate: \(E_0\cdot\exp(-(2\alpha/A^{(II)})(\sqrt K - \sqrt{K_0}))\)
Type II convergence rate
Let \(\omega^*\) minimize the time-integrated MSE \(I(\omega) = \int_0^T \mathbb{E}\!\left[\,(B^H_t - \hat B^H_t)^2\right] dt\) on geometric nodes \(\gamma_k = \gamma_{\min}^{1-z_k}\,\gamma_{\max}^{z_k}\), \(z_k = (k-1)/(K-1)\), \(k=1,\dots,K\). With endpoints scaled per-point as
\[ \gamma_{\min} \asymp \exp\!\Big(\!-\frac{\alpha\sqrt K}{(d - H)\,A^{(II)}}\Big), \qquad \gamma_{\max} \asymp \exp\!\Big(\!+\frac{\alpha\sqrt K}{H\,A^{(II)}}\Big), \]where \(\alpha = 1.06418\) (Bayer & Breneis 2023) and
\[ A^{(II)}(H) = \sqrt{\frac{1}{H} + \frac{1}{d - H}}, \qquad d = \begin{cases} 3/2 & H < 1/2 \quad (\text{rough, } \mu\text{-density}) \\ 5/2 & H > 1/2 \quad (\text{smooth, } \nu\text{-density}) \end{cases} \]the error obeys
\[ \mathcal{E}^{(II)}(\omega^*) \;\le\; C(H, T)\,\exp\!\Big(\!-\frac{2\alpha}{A^{(II)}}\sqrt K\Big). \]
Proof sketch (H < 1/2, rough regime).
The optimal weights \(\omega^*\) solve the linear system \(A\omega^* = b\), exactly the minimizer of the quadratic \(I(\omega) = \omega^\top A\omega - 2 b^\top \omega + c\). By quasi-optimality (Céa-style), \(I(\omega^*) \le I(\omega^{\text{cmp}})\) for any comparison weights on the same geometric nodes — so it suffices to exhibit good comparison weights. Taking Bayer & Breneis-style interpolatory weights on a head/middle/tail decomposition of the \(\mu\)-density Laplace representation \(G(u) = c_H \int_0^\infty e^{-\gamma u}\gamma^{-H-1/2}d\gamma\) (thesis Eq. 3.16), with blocks of size \(m \sim \sqrt K\) on the middle interval, gives head \(\sim \gamma_{\min}^{3-2H}\), tail \(\sim \gamma_{\max}^{-2H}\), middle \(\sim e^{-c\sqrt K}\). Balancing them yields \(A^{(II,<)}(H) = \sqrt{1/H + 1/(3/2 - H)}\).
Proof sketch (H > 1/2, smooth regime).
For H > 1/2 the natural representation switches to the \(\nu\)-density (thesis Eq. 3.17) with \(\tau\,\mathcal{L}(\nu)(\tau) = \tau^{H-1/2}/\Gamma(H+1/2)\) (Eq. 3.26), so \(G(u) = u \cdot K_\nu(u)\) where \(K_\nu(u) := \int_0^\infty e^{-\gamma u} \nu(\gamma)\,d\gamma\). The U-basis \(\hat B^{(II)}_H(t) = \sum_k \omega_k U_{\gamma_k}(t) = \int_0^t (t-s)\,\hat G_\nu(t-s)\,dW_s\) reduces fitting \(G\) to fitting \(K_\nu\) by a sum of exponentials \(\hat G_\nu(u)\). The Itô-isometry error gains an extra \(u^2\) factor inside:
\[I^{(II,>)}(\omega) = \int_0^T (T - u)\,u^2\,[K_\nu(u) - \hat G_\nu(u)]^2\,du.\]
Repeating the head/tail analysis on \(\nu(\gamma) \propto \gamma^{-(H-1/2)}\): head leading term is constant in \(u\) (absorbed by the \(\gamma_0 = 0\) Bayer–Breneis trick), residual is linear-in-\(u\) with prefactor \(\gamma_{\min}^{5/2 - H}\). After squaring and integrating against the \((T-u)\,u^2\) weight (the extra \(u^2\) pushes \(\int u^2 du\) up to \(\int u^4 du\)):
\[\text{head}_{II,>} \sim \frac{T^6}{30}\,\gamma_{\min}^{5 - 2H}, \qquad \text{tail}_{II,>} \sim \gamma_{\max}^{-2H}\;\text{(unchanged)}.\]
Balancing gives \(A^{(II,>)}(H) = \sqrt{1/H + 1/(5/2 - H)}\) and shifts the \((3/2 - H)\) coefficient in \(\gamma_{\min}\) to \((5/2 - H)\). The Y-basis approximation in
omega_optimized_2_mp realises this implicitly via finite-differencing of \(\partial_\gamma Y\) (thesis Sec. 3.7.4 / Eq. 3.31).
At H = 0.3: \(A^{(II)}\) = —, slope \(2\alpha/A^{(II)}\) = —.