Distribution dependent SDEs driven by additive continuous noise

Abstract

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

Distribution dependent SDEs driven by additive fractional Brownian motion

“…Step 1: weak existence. By hypothesis B : [0, T ]×R d ×P p (R d ) → R d is a uniformly continuous, bounded map; existence of weak solutions on [0, T ] then follows from [15,Proposition 3.10].…”

“…One classical way to show that the condition µ 0 ∈ L p ′ x is propagated at positive times, which has been exploited systematically after [12], is to impose boundedness of div b; in the setting of DDSDEs with general additive noise and regular drift b, an analogous statement can be found in [15,Proposition 4.3].…”

“…Gaussian processes satisfying a local non-determinism condition). In this sense, this work also serves as a comparison to the results from [15], where we explored in detail the DDSDE (1.1) in the opposite regime where no assumption whatsoever is imposed on W , thus no regularization can be observed.…”

McKean-Vlasov SDEs and SPDEs with Locally Monotone Coefficients

“…both the state and measure variables, whereas he need to impose the uniform boundedness on the coefficients. Afterwards, Galeati et al in [24] extended the results of [19] to the assumption of exponential integrability instead of the uniform boundedness. Recently, Li et al [43] established the strong convergence of Euler-Maruyama schemes for approximating distribution dependent SDEs by assuming globally Lipschitz w.r.t.…”

Distribution dependent SDEs driven by additive continuous noise

“…In this regard, another motivation for the current work is to provide a family of "baseline results" to be compared to the regularizing features of suitable nondegenerate choices of Y ; in our second article [28] we will study in detail the well-posedness of (1.1) for highly irregular B when the noise is sampled as a fractional Brownian motion.…”