Worst-Case Relative Growth of Functionally Generated Portfolios under Restricted Market-Weight Dynamics
Abstract
We study the worst-case relative growth rate of a fixed functionally generated portfolio under restrictions on market-weight dynamics over a compact convex market state space. For a nonempty compact forward shift-invariant path family, the pathwise wealth decomposition identifies the global rate with the minimum invariant average of the one-step L-divergence. For paths generated by a compact one-step transition relation, this rate is also the minimum over balanced transition measures, and a minimizing measure can be chosen on one relation basic set. For a fixed initial state, we study the lower limit of the minimum average L-divergence over the first t periods as t→∞. We represent this value using empirical transition laws generated from that state. Their common state marginals are supported on chain-recurrent states that are chain-accessible from the initial state. The lower bound is obtained by minimizing the average L-divergence over balanced transition measures on chain-accessible basic sets. Finite exact reachability gives an upper bound. Examples show that both bounds can be strict. We also give a finite exact reachability condition under which the minimum average L-divergence over the first t periods converges for paths confined to a basic set.
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BibTeX
@misc{xu2026worstcase,
title = {{Worst-Case Relative Growth of Functionally Generated Portfolios under Restricted Market-Weight Dynamics}},
author = {Jilong Xu and Xiaojun Cui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41603},
eprint = {2610.41603},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41603},
}
RIS
TY - UNPB
TI - Worst-Case Relative Growth of Functionally Generated Portfolios under Restricted Market-Weight Dynamics
AU - Jilong Xu
AU - Xiaojun Cui
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41603
UR - https://arxiv.org/abs/2610.41603
AB - We study the worst-case relative growth rate of a fixed functionally generated portfolio under restrictions on market-weight dynamics over a compact convex market state space. For a nonempty compact forward shift-invariant path family, the pathwise wealth decomposition identifies the global rate with the minimum invariant average of the one-step L-divergence. For paths generated by a compact one-step transition relation, this rate is also the minimum over balanced transition measures, and a minimizing measure can be chosen on one relation basic set. For a fixed initial state, we study the lower limit of the minimum average L-divergence over the first t periods as t→∞. We represent this value using empirical transition laws generated from that state. Their common state marginals are supported on chain-recurrent states that are chain-accessible from the initial state. The lower bound is obtained by minimizing the average L-divergence over balanced transition measures on chain-accessible basic sets. Finite exact reachability gives an upper bound. Examples show that both bounds can be strict. We also give a finite exact reachability condition under which the minimum average L-divergence over the first t periods converges for paths confined to a basic set.
ER -