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Register a new userDistribution dependent SDEs driven by additive continuous noise
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We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. 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.
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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 $Hin (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(cdot,mu) = fastmu(cdot) + g(cdot),quad f,gin B^alpha_{infty,infty}, quad alpha>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] 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.
In this paper, the existence and uniqueness of the distribution dependent SDEs with H{o}lder continuous drift driven by $alpha$-stable process is investigated. Moreover, by using Zvonkin type transformation, the convergence rate of Euler-Maruyama method is also obtained. The results cover the ones in the case of distribution independent SDEs.
We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness and regularity of the flow for the resulting equation, even when both the drift and the diffusion coefficients are distributional, thus extending the program of regularization by noise to the case of multiplicative SDEs. Our method relies on a combination of the non-linear Young formalism developed by Catellier and Gubinelli, and stochastic averaging estimates recently obtained by Hairer and Li.
In this paper we study the asymptotic properties of the power variations of stochastic processes of the type X=Y+L, where L is an alpha-stable Levy process, and Y a perturbation which satisfies some mild Lipschitz continuity assumptions. We establish local functional limit theorems for the power variation processes of X. In case X is a solution of a stochastic differential equation driven by L, these limit theorems provide estimators of the stability index alpha. They are applicable for instance to model fitting problems for paleo-climatic temperature time series taken from the Greenland ice core.
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorovs backward equations in some second order Sobolev spaces, which is analytically interesting in itself.
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