Singular Tracking of Fractional Brownian Motion under Proportional Costs

We study causal finite-variation controls that track fractional Brownian motion under quadratic tracking loss and proportional variation costs. Let V_λ(T;a,b,σ) be the optimal value on $[0,T]$ when the target is σ BH, the tracking weight is $a>0$, and the variation weight is λ b with $b>0$. For every H∈(0,1), there is a constant κH∈(0,∞) such that\[ V_λ(T;a,b,σ) ~ κ_H T a(1-H)/(1+H) (λ b)2H/(1+H) σ2/(1+H), λ↓0.\] The lower bound with exponent $2H/(1+H)$ also holds over noncausal finite-variation controls. The constant κH is the long-run value of a fractional cell problem. We prove that the same constant is obtained when the controller sees the full past before time zero. We also show that κ1/2=12(32)2/3, give primal and dual bounds on κH, and prove a local homogenization theorem for targets with a filtered fractional tangent. This gives the limit for a class of Gaussian Volterra targets. For $H<1/2$, we also prove convergence of finite-dimensional Markov cell values in a joint factor-resolution and long-horizon limit.