Singular Tracking of Fractional Brownian Motion under Proportional Costs

Abstract

We study adapted finite-variation tracking of fractional Brownian motion under quadratic loss and proportional adjustment costs. For $H\in(0,1)$, \[ V_λ(T;a,b,σ)\underset{λ\downarrow0}{\sim}κ_HT a^{(1-H)/(1+H)}(λb)^{2H/(1+H)}σ^{2/(1+H)}. \] The positive long-run cell value $κ_H$ is unchanged by full past information. A bounded full-past reset correction yields finite-horizon bounds and continuity in $H$. The Brownian value is $κ_{1/2}=\frac12(\frac32)^{2/3}$. Coupling local paths with their filtrations gives a local limit, which we verify for Gaussian Volterra targets. Every asymptotically optimal control has the same leading tracking--variation cost split. Primal and dual bounds quantify the cell value. For $H<1/2$, we construct a positive-factor approximation and relate factor resolution to horizon length.