A stale OU price can turn a large mean-reversion bet into a much smaller one. In Dolinsky's example, half a time unit of observation delay cuts the certainty equivalent over a ten-unit horizon from 25 to about 2.975. The parameters are κ = σ = 1, H = 0.5 and T = 10; the price begins at its long-run mean. The investors differ only in when they see the price. For a trader, the sharper warning concerns a tempting shortcut: sizing from the forecast price produces infinite expected exponential loss past an explicit horizon.
Delay changes the value curve
One asset follows dr = κ(θ - r)dt + σdW. Cash pays nothing, trades have no friction, positions have no limits, and κ, θ and σ are known. The investor maximizes exponential utility while observing prices H time units late and trading at today's price.
Mean reversion supplies the return. With full information, the position is proportional to θ - r_t and scales with 1 + κ(T - t); more time left means a larger bet. Guasoni, Nagy and Rásonyi showed that its certainty equivalent is νT + κ²T²/4, where ν = κ²(θ - x)²/(2σ²) measures the starting distance from the mean. The T² term gives full information quadratic growth. Our earlier note on weakly mean-reverting prices, /articles/exponential-investors-with-weakly-mean-reverting-prices, covered that horizon-scaled trade.
Dolinsky's closed-form delayed solution grows more slowly. For fixed H and increasing T, the certainty equivalent is (ν + λH)T - pH log T + O(1), where λH = κ/(e^{2κH} - 1). Delay matters relative to the speed of mean reversion: at κH = 0.10 (0.144 half-lives), λH/κ is 4.517; at κH = 1.00, it is 0.157; at κH = 2.00, it is 0.019. As the author puts it, "the cost of delay depends on the time scale of the signal, rather than on the delay measured in calendar units." Shrinking H recovers full-information value at a fixed horizon. Holding any H above zero leaves growth linear as T increases.
The two limits do not commute.
Observations play no part in earning the νT term. A deterministic position, κ(θ - x)[1 + κ(T - t)]/σ², earns it; that is also the optimal delayed position until the first price observation arrives at time H. The absolute cost of delay is independent of both the starting displacement and σ. In the worked example, the cost stays 22.025 whether the price starts at its mean or one unit away; the corresponding values are 7.975 and 30.
This is a theory paper, with no market data, estimation or trading results and hence no empirical claim. Every number above comes from a closed form or a single simulated path. The ETF figures below are ours alone.
What goes wrong with the forecast price?
A desk might retain the full-information rule and substitute the conditional mean of today's price for the unseen price. For this OU process, the substitution shrinks the last observed deviation by e^{-κH}. Dolinsky finds that the resulting projected rule is admissible and has finite expected squared position. Yet its expected exponential loss is infinite at every horizon at or beyond T_H = 2H + e^{κH}(2H/(3π) + 3π/(2κ²H)).
The proof picks out a Gaussian direction in the gain whose quadratic coefficient reaches -1/2 or lower. The bound, the author writes, "is sufficient and need not be the first divergence horizon." It establishes divergence from that point onward without establishing finiteness at shorter horizons.
The unseen window generates P&L that enters the exponential loss. Forecasting the price leaves that cost unpriced. Dolinsky's optimizer retains the deterministic position and subtracts a history term Y, which solves a linear Volterra equation driven by observed prices. Today's Y is a horizon-dependent coefficient C_t times the sum of the observed move r_{t-H} - x and a cosh-weighted integral of Y over the last H units. The position therefore uses the observed path, even though the price itself is Markov. Delayed control carries memory.
The choices our build had to make
The paper specifies its numerics with seed 20261011, κ = 1 and grid Δ = 1/1600. Against that grid, the refinement differences are 0.01551, 0.00897 and 0.00244 for Δ = 1/200, 1/400 and 1/800. The author calls them refinement comparisons rather than error bounds. Position constraints, transaction costs and parameter uncertainty are also outside the model. An implementation has to choose how to handle each.
Horizon dominates those choices here. We used a one-session delay and 20-session episodes, selecting ETFs with fitted half-lives of 3 to 60 sessions. This gives κH between about 0.012 and 0.23, or 0.017 to 0.33 half-lives, and κT between 0.23 and 4.6. Across that same κH range, the paper's bound puts κT_H near 26 at the fast end and above 400 at the slow end. Our episodes ended well before the bound applied to the projected rule. They were also far from the long-run regime of the value formula.
Our 10% position-value cap matters separately: it removes the unbounded-position channel through which divergence occurs. A daily step is coarse in the paper's units, too, with κΔ between 0.012 and 0.23 versus 0.000625 on its simulated path. The paper sets risk aversion to 1 and scales positions by 1/α; we chose α = 0.0001 per dollar. Its parameters are known. We estimated ours on 504 sessions, then held them fixed for each 20-session episode.
Our ETF build lost 0.60% from 2020 to mid-2024
These are our backtest figures for 2020-01-01 to 2024-07-01, annualized from 2020-02-04. Each year's universe comprised its top 50 US ETFs by volume, subject to median daily dollar volume above $20m, an ADF p-value of 0.05 or lower, and stable κ across half-windows. Signals used closes through t-1; fills were at the close of t. We charged $0.004 a share, with a $1 minimum. After those commissions, the Volterra book returned -0.60%. Its Sharpe was -0.06, win rate 46.22%, profit factor 0.96 and maximum drawdown 3.86% across 1,351 trades. The paper reports no trading performance, so it supplies no author figure for comparison.
The book took little risk: annualized volatility was 2.27%, beta to SPY was 0.02, and drawdown was 3.86%. Our α and the cap kept positions small. We omitted market impact, shorting costs and financing costs for up to 4x leverage; each would make the -0.60% result worse. We have not established when each year's volume ranking became known, leaving possible selection look-ahead in the universe. We tracked the projected rule in a separate book but have no paired result to report.
This result belongs to our build. The likeliest explanations are parameter estimates in place of the paper's known parameters, with the ADF screen run on the same 504 sessions used for fitting, and 20-session episodes too short to reach either the long-horizon asymptotics or the divergence bound. Since κT never exceeded 4.6, this run cannot distinguish linear from quadratic growth.
A longer test
Testing divergence would require uncapped positions and episodes long enough for κT to clear roughly 26 even for the fastest names. That threshold is only a sufficient bound. The author cautions that the single plotted path "does not estimate expected utility"; finite samples cannot establish finiteness of an exponential moment either.
If an uncapped projected book showed no widening of its worst losses as horizons passed κT of 26, I would treat the divergence as a tail risk manageable with position limits and keep the rule.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.