This paper studies how a specific kind of mean reversion in prices affects what a risk‑averse investor can earn over long horizons. The authors consider a simple continuous‑time market where the risky price has a steady linear drift plus a mean‑reverting fluctuation. The mean reversion is “weak,” meaning it pulls the price back with a force proportional to |x|^β with β between 0 and 1. For an investor who maximizes exponential utility (a common model of risk aversion), they show the certainty equivalent grows on the order of T^{2β+1} as the time horizon T becomes large.

Quick definitions: exponential utility means the investor seeks to maximize the expected value of exp(−wealth), which penalizes uncertainty. The certainty equivalent is the sure amount of money that gives the same utility as the risky outcome — it is a convenient single number summarizing an investor’s payoff. The main finding is an asymptotic rate: no admissible trading strategy can make the certainty equivalent grow faster than a constant times T^{2β+1}, and the authors also give an explicit family of strategies that attains this same growth order.

On the modeling side, the risky price S_t is written as a deterministic linear trend plus a mean‑reverting term X_t. The process X_t follows a stochastic differential equation with drift −sgn(X_t)|X_t|^β and a Brownian noise term. The paper fixes the riskless rate at zero and studies investors with risk‑aversion parameter equal to one. The analysis is asymptotic in the horizon T → ∞: the claimed growth concerns how the certainty equivalent scales for very long investment horizons.

How they prove the result is twofold. For the upper bound (showing no strategy can do better than order T^{2β+1}), they use an entropy duality argument. This introduces a change of probability measure that turns the price into a martingale, and the entropy of that change of measure controls the maximal exponential utility. Estimating that entropy leads directly to the T^{2β+1} upper bound. For the matching lower bound (showing the order is attainable), they construct an explicit family of trading rules. Each rule has the form H_t = (T+1−t)^{-1}·φ_ε(S_t), where φ_ε is a smoothed version of the power function sgn(x)|x|^β. The smoothing is needed because the raw power function is not regular at the origin. The authors prove the constructed strategy is admissible, control its terminal wealth, and check exponential moment bounds needed to show it reaches the same T^{2β+1} growth.