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Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion
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abstract
We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $\sigma$ that is at least $\gamma$-H\"older continuous for $\gamma > \frac{1}{2H} \vee \frac{1-H}{H}$. This improves upon the long-standing results of [Lyo94 , Lyo98 , Dav08] which cover the same regime but require $\sigma$ to be at least $\frac{1}{H}$-H\"older continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to [L\^e20, Ger22, MP22].
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On the density of singular SDEs with fractional noise and applications to McKean-Vlasov equations
Singular fBm-driven SDEs with distributional drifts have densities with Besov regularity and Gaussian tails, and the associated McKean-Vlasov equations are well-posed down to the subcritical scaling threshold.
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