Pathwise uniqueness for multiplicative young and rough differential equations driven by fractional Brownian motion

Toyomu Matsuda; Avi Mayorcas

doi:10.1214/25-AAP2200

Pathwise uniqueness for multiplicative young and rough differential equations driven by fractional Brownian motion

Toyomu Matsuda
, Avi Mayorcas

Independent

Research output: Contribution to journal › Article › peer-review

Abstract

We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter H∈(1/3,1) with volatility coefficient σ that is at least γ-Hölder continuous for Formula Presented. This improves upon the long-standing results of (Math. Res. Lett. 1 (1994) 451–464; Rev. Mat. Iberoam. 14 (1998) 215–310; Appl. Math. Res. Express. AMRX (2008) Art. ID abm009) which cover the same regime but require σ to be at least 1/H-Hölder continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to (Electron. J. Probab. 25 (2020) Paper No. 38; Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023) 714–729; Forum Math. Sigma 12 (2024) Paper No. e52).

Original language	English
Pages (from-to)	3441-3489
Number of pages	49
Journal	Annals of Applied Probability
Volume	35
Issue number	5
Early online date	20 Oct 2025
DOIs	https://doi.org/10.1214/25-AAP2200
Publication status	Published - 31 Oct 2025

Acknowledgements

Both authors wish to thank K. Dareiotis, L. Galeati, M. Gerescér and N. Perkowski for illuminating discussions during the course of this project. The main portion of this research was conducted when TM was a Ph.D. student at Freie Universität Berlin.

Funding

TM was supported by the German Science Foundation (DFG) via the IRTG 2544. AM gratefully acknowledges financial support from DFG Research Unit FOR2402.

Keywords

fractional Brownian motion
regularisation by noise
rough paths
Stochastic differential equations
stochastic sewing

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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https://arxiv.org/abs/2312.06473Licence: CC BY

Cite this

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title = "Pathwise uniqueness for multiplicative young and rough differential equations driven by fractional Brownian motion",

abstract = "We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter H∈(1/3,1) with volatility coefficient σ that is at least γ-H{\"o}lder continuous for Formula Presented. This improves upon the long-standing results of (Math. Res. Lett. 1 (1994) 451–464; Rev. Mat. Iberoam. 14 (1998) 215–310; Appl. Math. Res. Express. AMRX (2008) Art. ID abm009) which cover the same regime but require σ to be at least 1/H-H{\"o}lder continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to (Electron. J. Probab. 25 (2020) Paper No. 38; Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023) 714–729; Forum Math. Sigma 12 (2024) Paper No. e52).",

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N2 - We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter H∈(1/3,1) with volatility coefficient σ that is at least γ-Hölder continuous for Formula Presented. This improves upon the long-standing results of (Math. Res. Lett. 1 (1994) 451–464; Rev. Mat. Iberoam. 14 (1998) 215–310; Appl. Math. Res. Express. AMRX (2008) Art. ID abm009) which cover the same regime but require σ to be at least 1/H-Hölder continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to (Electron. J. Probab. 25 (2020) Paper No. 38; Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023) 714–729; Forum Math. Sigma 12 (2024) Paper No. e52).

AB - We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter H∈(1/3,1) with volatility coefficient σ that is at least γ-Hölder continuous for Formula Presented. This improves upon the long-standing results of (Math. Res. Lett. 1 (1994) 451–464; Rev. Mat. Iberoam. 14 (1998) 215–310; Appl. Math. Res. Express. AMRX (2008) Art. ID abm009) which cover the same regime but require σ to be at least 1/H-Hölder continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to (Electron. J. Probab. 25 (2020) Paper No. 38; Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023) 714–729; Forum Math. Sigma 12 (2024) Paper No. e52).

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Pathwise uniqueness for multiplicative young and rough differential equations driven by fractional Brownian motion

Abstract

Acknowledgements

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Keywords

ASJC Scopus subject areas

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