Distribution dependent SDEs driven by additive fractional Brownian motion

By Lucio Galeati, Fabian A. Harang, Avi Mayorcas. Probability theory and related fields, 2021

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We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $$H\in (0,1)$$ H ∈ ( 0 , 1 ) . We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$\begin{aligned} B(\cdot ,\mu )=(f*\mu )(\cdot ) + g(\cdot ), \quad f,\,g\in B^\alpha _{\infty ,\infty },\quad \alpha >1-\frac{1}{2H}, \end{aligned}$$ B ( · , μ ) = ( f ∗ μ ) ( · ) + g ( · ) , f , g ∈ B ∞ , ∞ α , α > 1 - 1 2 H , thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.

Subfields: Applied Mathematics, Computational Mechanics, Finance

View Paper: https://www.semanticscholar.org/paper/2b913b8d13d80fdcde1c8b04059bd12105154099