Well-posedness of distribution dependent SDEs with singular drifts

By Michael Röckner, Xicheng Zhang. Bernoulli, 2018

149 citations recorded by the source corpus.

In this paper we consider the following distribution dependent SDE: $$ {\mathrm d} X_t=\sigma_t(X_t,\mu_{X_t}){\mathrm d} W_t+b_t(X_t,\mu_{X_t}){\mathrm d} t, $$ where $\mu_{X_t}$ stands for the distribution of $X_t$. We show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in $\mu$ about $b$ and $\sigma$. In particular, we extend the results of Krylov-R\"ockner to the distribution dependent case.

Subfields: Applied Mathematics, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/3f66c3b84e5807b1dbf5a56a06e850c0605d5ba8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Economics and Econometrics

In Finance

Citations in the corpus

Strong convergence order for slow–fast McKean–Vlasov stochastic differential equations (2019, background): “, [30, 37]), and see [35] for the case of singular drifts.”
Distribution dependent SDEs driven by additive fractional Brownian motion (2021, methodology): “This also partially answers the ongoing debate from [38, 24, 25] on whether the drift B should be taken Lipschitz continuous in the measure argument μ w.”
Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance (2020, background): “ (1.4) bt(x,γ) := Z Rd ˜b t(x,y)γ(dy), where ˜b t(x,y) has linear growth in x uniformly in t and y. Since ˜bt(x,y) is bounded in y, bt(x,·) is Lipschtiz continuous in the total variation distance W0. [19] considers the same type drift and proves the well-posedness of (1.1) under the conditions that ”