Optimal Time-Dependent Jump Truncation for Time-Singular Lévy Processes
Abstract
We study optimal jump truncation for the additive time-singular pure-jump model of the form XT=∫0T t−σ dZt, where Z is a Lévy process and σ≥ 0. For a fixed expected jump cost, we minimize the residual small-jump variance over measurable time-dependent cutoffs. Under regularity and tail assumptions on the Lévy measure, we prove that the problem admits an optimal cutoff, unique up to a.e. equality, of the form r∗(t)=(c∗ tσ)∧ 1. In the symmetric α-stable case, we obtain explicit matched-cost comparisons with the classical fixed cutoff and show that within the admissible non-truncated regime the Dynamic Cutting family is strictly better than the classical fixed cutoff whenever 0<σ<1/2. Finally, for symmetric Lévy measures and cutoffs satisfying the corresponding Lp-integrability assumptions, we derive the weak-error bound Wp(r)≤ Cp ℰp/2(r), 0<p<2, demonstrating that any cutoff which minimizes the residual variance also minimizes the corresponding upper bound for the weak error.
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BibTeX
@misc{knopova2026optimal,
title = {{Optimal Time-Dependent Jump Truncation for Time-Singular Lévy Processes}},
author = {Victoria Knopova and Denis Platonov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15343},
eprint = {2609.15343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15343},
}
RIS
TY - UNPB
TI - Optimal Time-Dependent Jump Truncation for Time-Singular Lévy Processes
AU - Victoria Knopova
AU - Denis Platonov
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15343
UR - https://arxiv.org/abs/2609.15343
AB - We study optimal jump truncation for the additive time-singular pure-jump model of the form X_T=∫_0^T t^−σ dZ_t, where Z is a Lévy process and σ≥ 0. For a fixed expected jump cost, we minimize the residual small-jump variance over measurable time-dependent cutoffs. Under regularity and tail assumptions on the Lévy measure, we prove that the problem admits an optimal cutoff, unique up to a.e. equality, of the form r^∗(t)=(c^∗ t^σ)∧ 1. In the symmetric α-stable case, we obtain explicit matched-cost comparisons with the classical fixed cutoff and show that within the admissible non-truncated regime the Dynamic Cutting family is strictly better than the classical fixed cutoff whenever 0<σ<1/2. Finally, for symmetric Lévy measures and cutoffs satisfying the corresponding L^p-integrability assumptions, we derive the weak-error bound W_p(r)≤ C_p ℰ^p/2(r), 0<p<2, demonstrating that any cutoff which minimizes the residual variance also minimizes the corresponding upper bound for the weak error.
ER -