Dolinsky's frictionless gain grows like n, while the turnover suggested by the optimal positions grows like n^{1+H}. The paper computes the asymptotic growth rate of the normalized certainty equivalent, -(1/n) log inf E[exp(-V_1)], for high-frequency trading in a discretized fractional Brownian motion market. Rescaled by n^{-H}, the large, jumpy positions converge in finite-dimensional distributions to a white-noise-type Gaussian field. Successive holdings then resemble independent draws. Dolinsky summarizes the result directly: "in the high-frequency limit, the dependence between optimal positions at different trading times disappears."

Combining those exponents produces turnover of order n^{1+H} for a prize of order n. This inference is mine. Dolinsky does not make it, and Theorem 1.2 establishes finite-dimensional convergence without tightness, so it yields no turnover functional. Nor does the paper claim that the strategy is tradable. Transaction cost, spread and impact terms are absent because Dolinsky asks a different question.

The objection is economic. The mathematics holds.

The market substitution also needs to be explicit. Dolinsky studies the synthetic price process S = B^H, which has no traded market behind it. We would transfer its return-history mechanism to liquid US equities and ETFs using minute bars. The model treats price as fractional Brownian motion with a known Hurst parameter and permits frictionless trading on a discrete grid. Real equity and ETF returns will violate those assumptions, while H would require estimation. Its high-frequency asymptotics allow rapidly varying and potentially unbounded positions. A live book must operate within leverage, turnover, liquidity and cost limits. Minute OHLCV bars allow one-minute implementation, yet they reveal neither bid-ask spreads nor queue position or true execution cost.

A driftless, costless market

The risky asset is fractional Brownian motion itself, S = B^H. Beside it is a savings account that pays nothing. Trades occur on {0, T/n,..., T}. Self-similarity lets Dolinsky set T = 1, while invariance of the exponential-utility problem under scaling the position vector lets him set risk aversion to alpha = 1. Both are exact model normalizations.

There is no drift or risk premium. The asset has zero mean and may become negative. Investor gains come entirely from the autocovariance of fractional Gaussian noise, whose increments X_j satisfy rho_H(k) = (1/2)(|k+1|^{2H} - 2|k|^{2H} + |k-1|^{2H}). Correlation is positive when H is above 1/2, negative below 1/2 and absent at 1/2.

Discrete trading keeps the problem well posed. The paper observes that fBm ceases to be a semimartingale away from H = 1/2, adding that "if trading is restricted to a discrete trading grid, arbitrage opportunities disappear."

Dolinsky and Zuk (2023) provide the finite-n value. The quantity inf E[exp(-V_1)] equals the product of det Gamma_n^H and the diagonal precision entries, raised to minus one half. Gamma_n^H denotes the covariance matrix of the n increments. For its determinant, Dolinsky uses the first Szegő limit theorem in the form supplied by Guo, Li and Zhou (2024).

Each diagonal precision entry is 1/v_{i-1,n-i}. The term v is the error from interpolating one increment using p neighbours to the left and q to the right. Kolmogorov and Salehi's interpolation formula gives the limiting error. The remaining argument squeezes the n - 2m interior indices through v_infinity <= v_{p,q} <= Var(X_0) = 1.

The resulting expression deserves attention. Its growth rate is (1/2) log of the arithmetic mean divided by the geometric mean of 1/f_H, the inverse spectral density. AM-GM makes the rate non-negative. Equality occurs exactly for a flat spectral density, the Brownian case. The edge therefore comes from spectral non-flatness and remains positive on both sides of H = 1/2. Figure 1 plots the limit against H, although the accompanying text gives no table of values.

What the position rule actually does

The second theorem contains the paper's tradable content. More precisely, the useful object is the limit of the position rule. Corollary 1.3(I) in Dolinsky and Zuk writes the optimal portfolio explicitly. At date (i-1)/n, the position rescaled by n^{-H} equals minus the sum over j < i of (Lambda_n^H)_{ij} X_j.

This is a linear filter of realized increments. Its weights come from a row of the precision matrix. Dolinsky identifies that row as the f_H-weighted projection of 1/f_H onto the frequencies available at the time. Away from the boundaries, it converges to the centered stationary Gaussian sequence Y_m = minus the sum over k >= 1 of c_k^H X_{m-k}, with c_k^H = (1/2pi) integral e^{iku}/f_H(u) du.

In trading terms, the rule is a fixed-coefficient moving average of past returns, built by inverting the model's spectral density f_H. Nothing exotic.

The proof pays for that simple interpretation. Projection onto the negative-frequency Fourier modes is orthogonal in unweighted L2. Orthogonality disappears under the f_H-weighted inner product, and boundedness then requires f_H to belong to the Muckenhoupt A_2 class. It does. Near zero, f_H behaves like |lambda|^{1-2H}; the exponent 1-2H remains in (-1,1) for every H in (0,1). The needed theorem comes from Hunt, Muckenhoupt and Wheeden. Dolinsky explicitly warns that ordinary Hilbert-space theory does not supply this step automatically.

Why faster trading works against the result

The paper leaves the following arithmetic undone. The certainty equivalent scales as n times the AM/GM rate. Meanwhile, convergence of the n^{-H}-rescaled position to a limit with variance sigma_H^2 implies raw-position variance of order n^{2H} sigma_H^2. In the finite-dimensional limit, consecutive rescaled positions become asymptotically independent.

Speeding up therefore brings no smoothing to the position path. Adding n independent jumps of order n^H suggests turnover of order n^{1+H}. Proportional costs on shares traded would then rise superlinearly while the prize rises linearly, sending the cost-to-gain ratio upward like n^H.

Within the model, faster trading always wins.

Two qualifications matter. Theorem 1.2 proves finite-dimensional convergence, with no tightness proven. Its limit cannot support a turnover functional, which leaves my scaling as a heuristic combination of the two exponents. The divergence also weakens as H approaches zero, since n^{1+H} moves toward n. For H > 1/2 the difference is harder to dismiss, and this is the region motivating the paper: fBm "can display the long-range dependence observed in empirical data when H > 1/2."

The whitening result expresses the same issue through positions. Long-range dependence in returns disappears from the optimal book in the high-frequency limit because the filter inverts the spectrum. Dolinsky's wording is exact: "the dependence between optimal positions at different trading times disappears." Successive holdings that resemble independent draws offer no netting.

We have not traded this. Implementing the optimal rule requires the exact precision matrix and exact Fourier coefficients of 1/f_H. Both would need to be estimated outside the model. We expect bid-ask bounce to dominate measured antipersistence at one-minute frequency, and bounce appears in the quote rather than in an executable sale price.

A finite-n version of Theorem 1.1 containing a proportional cost term would change my view. The paper supplies no such result. Its conclusions are asymptotic in n, without finite-n rates or a cost term. The AM/GM formula correctly measures the value of dependence, but the quantity it measures remains unpriced.