Sharp Weak-Trace Regularity of Stochastic Cross Integrals
Abstract
We prove a sharp spatial regularity theorem for the Itô and Stratonovich cross integrals of continuous semimartingales in separable Hilbert spaces. Without global moment assumptions, both integrals admit versions that are almost surely jointly continuous in the two time variables, with values in the little weak-trace ideal. This regularity is optimal even for Wiener processes with deterministic trace-one covariance. Given any positive sequence tending to zero, we construct such a process for which the singular values of its Lévy area, multiplied by their indices, tend to zero but have termwise ratios to the prescribed sequence tending to infinity almost surely. The same property holds simultaneously for both self-integrals. The upper bound follows from a rank-one cancellation in Itô's formula for a determinant, together with finite-rank approximation. For the lower bound, conditional Gaussian estimates and orthogonal time functions yield finite-dimensional spectral bounds, which are combined through a trace-one covariance to obtain lower bounds for all sufficiently large indices.
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BibTeX
@misc{zhao2026sharp,
title = {{Sharp Weak-Trace Regularity of Stochastic Cross Integrals}},
author = {Guangqian Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.08756},
eprint = {2608.08756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08756},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Weak-Trace Regularity of Stochastic Cross Integrals
AU - Guangqian Zhao
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2608.08756
UR - https://arxiv.org/abs/2608.08756
AB - We prove a sharp spatial regularity theorem for the Itô and Stratonovich cross integrals of continuous semimartingales in separable Hilbert spaces. Without global moment assumptions, both integrals admit versions that are almost surely jointly continuous in the two time variables, with values in the little weak-trace ideal. This regularity is optimal even for Wiener processes with deterministic trace-one covariance. Given any positive sequence tending to zero, we construct such a process for which the singular values of its Lévy area, multiplied by their indices, tend to zero but have termwise ratios to the prescribed sequence tending to infinity almost surely. The same property holds simultaneously for both self-integrals. The upper bound follows from a rank-one cancellation in Itô's formula for a determinant, together with finite-rank approximation. For the lower bound, conditional Gaussian estimates and orthogonal time functions yield finite-dimensional spectral bounds, which are combined through a trace-one covariance to obtain lower bounds for all sufficiently large indices.
ER -