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∈(0,1), Vλ(T;a,b,σ)undersetλ↓0∼κHT a(1−H)/(1+H)(λb)2H/(1+H)σ2/(1+H). The positive long-run cell value κH is unchanged by full past information. A bounded full-past reset correction yields finite-horizon bounds and continuity in H. The Brownian value is κ1/2=1/2(3/2)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.
Related
- Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion
- Fractional Brownian motion with multivariate time: persistence exponents
- Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion via Fractional Calculus Techniques
- The Numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion
- Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
BibTeX
@misc{autran2026singular,
title = {{Singular Tracking of Fractional Brownian Motion under Proportional Costs}},
author = {Alexandre Autran},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01727},
eprint = {2611.01727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01727},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Singular Tracking of Fractional Brownian Motion under Proportional Costs
AU - Alexandre Autran
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01727
UR - https://arxiv.org/abs/2611.01727
AB - We study adapted finite-variation tracking of fractional Brownian motion under quadratic loss and proportional adjustment costs. For H∈(0,1), V_λ(T;a,b,σ)undersetλ↓0∼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=1/2(3/2)^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.
ER -