The trade reduces to a gradient: hold $\hat\theta_t = \nabla\hat\varphi(X_t)$, with $\hat\varphi$ fixed on the price space before trading begins.
Neither wealth nor the volatility factor becomes a state variable. Drift estimates also disappear. Itkin, Koch, Larsson and Teichmann prove that this rule achieves the worst-case long-run growth rate across every law where $X$ is a semimartingale, its covariance averages to $A$, its time averages converge to $p$, and the laws of $X_t$ are tight. The drift of $X$ and all dynamics of $Y$ remain free. The authors put the surprising result plainly: the optimal strategy "does not depend on the factor process $Y$".
We are building an approximation with a small universe, dimension reduction on the factor, a regularised density estimate from finite daily data and a numerical PDE solution. Nothing produced by that exercise tests the paper's asymptotic guarantee.
The two required inputs
The construction starts with $c_X(x,y)$, the instantaneous covariance of the $d$ assets as a function of prices $x$ and a factor $y$ in a set $D \subseteq \mathbb{R}^m$. It also needs $p(x,y)$, the joint invariant density of prices and factor. Those are the only inputs. Because instantaneous drift cannot be observed pathwise at equity noise levels, the admissible class includes every drift compatible with the specified covariance and ergodic behaviour.
The two objects collapse immediately into
$A(x) = \int_D c_X(x,y)\,p(x,y)\,dy.$
Section 6.2 gives a particularly readable case. Write $\bar p(x)$ for the marginal density of $X$ and set $\bar c_X(x) = \mathbb{E}[c_X(X,Y)\,|\,X=x]$. Then $A(x) = \bar c_X(x)\bar p(x)$, the conditional-average covariance weighted by the frequency with which prices occupy $x$.
For a functionally generated portfolio $\theta = \nabla\varphi(X)$, Itô, the ergodic constraint and tightness of the laws of $X_t$ eliminate the boundary term $\varphi(X_T)/T$. The remaining growth rate is $-\tfrac12\int_E \mathrm{tr}(A\nabla^2 e^{\varphi})/e^{\varphi}$. Every admissible measure gives the same number. This invariance drives the paper's result, supplying a performance guarantee without a return forecast and meeting the observability property Fernholz requested in Problems 3.1.7 and 3.1.8.
Optimising over $\varphi$ yields the Euler-Lagrange equation $\mathrm{div}(A\nabla\hat\varphi - \tfrac12\,\mathrm{div}\,A) = 0$. The attained rate is $\lambda = \tfrac12\int_E \nabla\hat\varphi^\top A\,\nabla\hat\varphi$. For the upper bound, the authors construct an adversarial measure under which $X$ has drift $\tilde c_X\nabla\hat\varphi$. Under that measure, $\nabla\hat\varphi$ is growth-optimal outright, leaving no strategy able to exceed it. Min and max coincide.
Why does the factor vanish?
Everything about the volatility factor that matters has already entered $A$. Factor models can differ in both $c_X$ and $p$, yet produce identical strategies and growth rates whenever their $p$-weighted averages agree. Section 5.2 goes further. Even after prescribing the entire joint covariation matrix of $X$ and $Y$, neither the off-diagonal block $c_{XY}$ nor the factor block $c_Y$ affects the strategy or rate, including through averages.
When $y \mapsto c_X(x,y)$ is invertible, $Y_t$ can be recovered from $X_t$ and the observed covariance. The paper still concludes that knowing the path of $Y$ "does not improve the performance in an adversarially chosen measure".
Section 6.1 explains the result, with the third reason carrying most of the weight. $Y$ is uninvestable, while the objective is asymptotic log growth. More importantly, the local drift of $Y$ remains unrestricted. The construction therefore has one free degree of freedom available to satisfy one constraint, namely that the invariant density equal $p$. The authors acknowledge the implication: once the drift of $Y$ is restricted, the optimal strategy may begin to depend on $Y$. They leave that question "for future research". The theorem consequently says as much about the breadth of the adversary's choices as it does about the portfolio rule.
From an estimated density to holdings
The practical sequence is simple to state: estimate $p$ and $c_X$, integrate out the factor to obtain $A$, solve an elliptic PDE and differentiate. The PDE is the difficult step. Section 7.1 acknowledges that the Euler-Lagrange calculation becomes numerically demanding as $d$ increases. A tractable exception appears in the gradient case. Whenever $A^{-1}\mathrm{div}\,A = \nabla h$, the answer is $\hat\varphi = h/2$, eliminating the PDE solve.
For $d = 1$, the optimiser is explicit: $\hat\varphi = \tfrac12\log A$. Its holding equals $\tfrac12\,\mathbb{E}[\partial_x \log(c_X p(X,Y))\,|\,X=x]$, evaluated under the law whose density is proportional to $c_X p$, rather than under $p$ itself.
Example 7.4 is the cleanest portfolio-manager version. Suppose $c_X$ depends only on the factor and $p$ factorises as $p_X(x)p_Y(y)$. The solution becomes $\hat\varphi = \tfrac12\log p_X$, so the volatility surface drops out of the strategy entirely. Its rate is $\lambda = \tfrac18 \int_E (\nabla \log p_X)^\top \bar A\, (\nabla \log p_X)\, p_X\,dx$, which is strictly positive whenever $p_X$ is not constant. Growth comes from the stationary distribution of prices. Covariance remains only through the constant matrix $\bar A$, which scales the result.
Where the guarantee ends
This is a theorem paper with no data. Both $p$ and $c_X$ are treated as known exactly. The paper supplies no estimator, convergence result or error analysis, leaving the practical difficulty intact: avoiding drift estimation requires a joint stationary density in $d+m$ dimensions.
Trading is frictionless and continuous. Wealth is $\mathcal{E}(\int \theta^\top dX)$, and $\Theta$ contains every predictable $X$-integrable process. Long-only constraints and leverage caps are absent. The consequence is visible in the Beta example in Section 7.5. With $q_1 = q_2 = 1$, it gives $\hat\theta_t = (a_1+b_1-1)/(2X_t) - (a_2+b_2-1)/(2(1-X_t)) + \dots$. The parameter restrictions imply $a_1+b_1 > 2$. Hence the coefficient on $1/X_t$ exceeds $1/2$, and the position diverges as $X_t \to 0$.
The objective is a $T \to \infty$ limit in probability. We did not find a statement giving either the convergence rate to $\lambda$ or finite-horizon dispersion around it.
The geometry also imposes tight conditions. Assumption 2.2 (i) requires $D$ to be bounded and convex. Under Assumption 4.1 (ii), repeated as Assumption 5.7 (ii), $\lambda_{\min}(c_Y)p$ must be a power of a concave function. This condition supplies the weighted Poincaré constant $\mathrm{diam}(U)/\pi$ used throughout the later results.
The $K$-modifications deserve the authors' description as technical. They alter $c_X$ outside a compact $K$, where covariance could not be estimated anyway, and show that the process spends only $\mu((E\setminus K)\times D) \le \varepsilon$ of its time there. Corollary 5.6 removes the modification when an integrability condition holds. For $m = 1$, with $A^{-1}\mathrm{div}\,A$ a gradient, the authors argue that their methods cannot improve that condition.
A further loose end appears in Section 3: the worst-case measure can be nonunique. Example 7.5 allows every pair $(\beta_1,\beta_2)$ satisfying $2-\alpha_i < \beta_i < \alpha_i$ to generate a worst-case measure, giving uncountably many. The value is attained, while the model attaining it remains unidentified.
Our test reaches the approximation
Testing the theorem itself would require exact knowledge of $p$ and $c_X$, followed by a solution of the variational problem on the true state space. We can instead run the estimated numerical approximation described above. The result provides evidence only about our approximation and leaves the asymptotic guarantee untouched.
Compared with Kardaras and Robertson's 2021 result, the advance is that the adversary may choose the factor dynamics as well, yet the answer remains unchanged. Evidence that $A$, when estimated from a realistic sample, is stable enough to keep $\nabla\hat\varphi$ from swinging across estimation windows would change my view of the method's practical value.