In the high-frequency limit, conditionally expected increments of fractional Brownian motion converge to a white noise, shedding their dependence on the path history and the forecasting horizon and making dynamic optimisation problems tractable. We find an explicit formula for locally mean–variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-adjusted profits are linear in the trading horizon and rise asymmetrically as the Hurst exponent departs from Brownian motion, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalise numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit.

在高频范围内，分数布朗运动的有条件预期增量会收敛为白噪声，从而消除了它们对路径历史和预测范围的依赖，并使动态优化问题易于解决。我们为局部均方差最优策略及其遵循分数布朗运动的资产价格的表现找到了一个明确的公式。没有交易成本，风险调整后的利润在交易范围内是线性的，并且随着赫斯特（Hurst）指数偏离布朗运动而非对称地上升，当指数达到零时保持有限，而在接近1时则偏离。交易成本会因短期信号而对大量投资组合更新造成不利影响，从而导致交易频率有限，可以选择有限的交易频率，以使交易成本的影响小到可观，