The theorem works; the trade remains unproven. Abi Jaber, Gutekunst, Herdegen and Hobson show that power-utility investment and consumption rules remain well defined and optimal even when volatility is rough enough to leave no instantaneous variance. For a desk, the arguable economic result comes down to a pair of plots with an allocation axis running from 50% to 60%.
Variance becomes the clock
Integrated variance U replaces spot volatility as the primitive. This continuous non-decreasing process records accumulated variance and serves as market time. The stock follows dS/S = r dt + Λ dU + ρ dM + sqrt(1−ρ²) dM⊥, with both martingales carrying quadratic variation U. Through Λ, the risk premium accumulates per unit of realized variance, making the instantaneous Sharpe scale with volatility.
Absolute continuity of U recovers standard stochastic-volatility models, including Heston. A singular U instead produces models such as hyper-rough Heston, where the kernel is fractional and h ∈ (−1/2, 0]. Integrated variance still exists there, though its derivative does not.
This construction covers the entire fractional range h ∈ (−1/2, 1/2). HJB methods fail throughout because the state is non-Markovian. The paper also delivers something for Heston users: in the classical constant-kernel case, the authors remove the Feller condition κθ ≥ ν²/2 required by their earlier work (Gutekunst, Herdegen and Hobson).
The main device is an infinite-horizon backward equation called an IVC-BSDE. Calendar time drives it because consumption occurs in dt, while the variance clock enters because risk arrives in dU. Everything follows from its solution F. Value equals X^{1−R}/(1−R)·F^{R/Rρ}, consumption divided by wealth equals F^{−1/Rρ}, and the risky weight equals Λ/R + (ρ/Rρ)·Z/F. The correlation-adjusted risk aversion is Rρ = (1−ρ²)R + ρ². Z, the BSDE's martingale loading, supplies the hedging term.
For existence and uniqueness, the authors place F between an ordered subsolution and supersolution, then apply monotone iteration. A weak-convergence argument gives stability. Duality completes the verification of optimality.
When does the theorem apply?
The first requirement is R > 1. Every main well-posedness result depends on it because the fixed-point map becomes monotone only in that range, a restriction the authors state directly.
The second is strong myopic well-posedness. Cumulative myopic consumption H must grow at least linearly. H collects the consumption rate an investor would choose if the market remained frozen in its current state. In the Volterra Heston application, H_t = η0·t + η1·U_t. Using the illustration's numbers (R = 2, δ = 0.03, r = 0.02, Λ = 1), the coefficients are η0 = 2.5% and η1 = 0.125. Once r, δ > 0 and R > 1, positivity follows automatically, so the condition imposes little in this setting.
The third issue arises in the incomplete market, where F has no formula. Verification instead rests on a sign. Malliavin calculus gives Z ≤ 0. The authors establish that result for regularised Volterra models and transfer it to the singular limit through their stability theorem. When ρ is negative, Z ≤ 0 produces positive hedging demand, lifting the stock position above the myopic Λ/R. The sign also bounds variance under the optimal measure and supplies the martingale property unavailable from Novikov. In the complete case, |ρ| = 1, an explicit solution follows from a Riccati-Volterra function ψ lying strictly between (κ̃ − sqrt(κ̃² + 2η1ν²))/ν² and 0.
A sharper supersolution for the incomplete case further requires κ̃ ≥ 0. Here κ̃ = κ + (R−1)/R·ρΛν, the mean-reversion speed under the paper's distorted measure Q. Optimality applies for any κ̃. Its sign affects only the precision of the numerical bracket.
Their illustration sets κ̃ = 0.195.
With κ = 0.3 and R = 2, a market where ρΛν falls below −0.6 (against −0.21 in the illustration) would take κ̃ below zero. Only the crude constant bound would then remain.
A narrow sensitivity exercise
The paper's two figures use R = 2, δ = 0.03, r = 0.02, Λ = 1, κ = 0.3, θ = 0.02, ν = 0.3, ρ = −0.7. The parameters come from an earlier Abi Jaber lifted-Heston table and are not estimated in this paper. The curves cover h = 0.5, 0.2, −0.2, −0.4 and −0.49, with Y0 ranging from 0.01 to 0.06, equivalent to 10% to 25% initial volatility.
Consumption looks nearly linear in Y0 on an axis spanning roughly 2.6% to 3.2%. Across the same range, the myopic rate η0 + η1·Y0 moves from about 2.6% to 3.25%. The optimal slope remains flatter than its myopic counterpart and grows steeper as h declines.
The practical claim lies in allocation. Myopic Λ/R is exactly 50% here. Optimal weights appear nearly flat in Y0 on a 50% to 60% axis, and they decline with h. Rougher volatility corresponds to a smaller stock position. Every plotted approximate weight remains on that 50% to 60% axis, limiting the hedge in these curves to about ten points above the 50% myopic weight. The roughness effect accounts for some fraction of those ten points, judged from a graph because the paper supplies no table of computed values.
The authors acknowledge the limitation of these allocation curves: "While it is generally not clear that these yield upper and lower bounds on the optimal portfolio, they should still provide a good approximation for the optimal portfolio." Their consumption bounds are rigorous brackets, described by the authors as "quite close to each other". The portfolio curves come from portfolios calculated using the two brackets, with no guarantee that the optimum falls between them. The paper offers no argument for "good approximation" beyond the preceding observation that the consumption brackets are close. That proximity provides a heuristic for the portfolio curves, without guaranteeing their accuracy.
The h-ordering deserves caution for a second reason. Across the curves, only h changes; κ, θ and ν retain their borrowed values. A desk fitting rough and hyper-rough specifications would recalibrate all three for each kernel. The ordering could change after that exercise. These figures answer a ceteris paribus question inside the model, while leaving open which kernel describes the market.
Frictions and the missing backtest
A live implementation faces two omitted problems: trading costs are absent, and Λ is treated as known exactly. Continuous rebalancing is assumed as well. With a hedge capped at roughly ten points of weight, estimation error in Λ and ρ could erase the edge. We did not find a calibration or out-of-sample exercise in the paper, consistent with its purpose as a theory paper.
We did not backtest the policy. Implementing it requires a numerical solution to the nonlinear infinite-horizon BSDE, while the incomplete-market numerics in the paper provide only brackets. Our data consist of discrete one-minute bars, which would require estimating the continuous clock U from intraday returns. Any run would therefore test a discretised, calibrated approximation rather than the authors' strategy.
A calibrated version would change my view: fit kernels by h to the same market, then report the resulting allocation gap. If the rough-versus-smooth difference remains above two percentage points after recalibration, the paper has supplied traders with a sizing input. For now, its contribution is a proof that such a rule exists.