We investigate the conditional McKean-Vlasov stochastic differential equations with jumps and Markovian regime-switching. We establish the strong wellposedness using L2-Wasser-stein distance on the Wasserstein space. Also, we establish the propagation of chaos for the associated mean-field interaction particle system with common noise and provide an explicit bound on the convergence rate. Furthermore, an averaging principle is established for two time-scale conditional McKean-Vlasov equations, where much attention is paid to the convergence of the conditional distribution term.
We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Meleard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Holder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamic…
In this paper, we show the strong well-posedness of Mckean-Vlasov SDEs with non-Lipschitz coefficients. Moreover, propagation of chaos and the convergence rate for Euler's scheme of Mckean-Vlasov SDEs are also obtained.
The existence and uniqueness are established for McKean-Vlasov SDEs driven by Lévy processes. By using an approximation technique and coupling by change of measures, Harnack inequalities are investigated for McKean-Vlasov SDEs driven by subordinate Brownian motions.
We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested by Cheridito et.al. In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike the earlier papers, the alternative formulation of this paper insists that the equation must hold under a non-dominated family of mutually singular probability measures. The key argument is a stochastic representation, suggested by the opt…
The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.
The Bismut formula is established for the intrinsic derivative of singular McKean-Vlasov SDEs, where the noise coefficient belongs to a local Sobolev space, and the drift contains a locally integrable time-space term as well as a time-space-distribution term Lipschitz continuous in the space and distribution variables. The results are new also for classical SDEs.
In this paper, the well-posedness for one-dimensional path dependent McKean-Vlasov SDEs with $α$($α\geq \frac{1}{2}$)-Hölder continuous diffusion is investigated. Moreover, the associated quantitative propagation of chaos in the sense of Wasserstein distance, total variation distance as well as relative entropy is studied.
One proves the uniqueness of distributional solutions to nonlinear Fokker--Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean--Vlasov stochastic differential equation (SDE).
By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present the exponential ergodicity in the relative entropy and (weighted) Wasserstein distances for reflecting McKean-Vlasov SDEs under the dissipative and partially dissipative conditions respectively. The main results are illustrated by non-symmetric singular granula…
The goal of this paper is to approximate several kinds of {\it Mckean-Vlasov SDEs} with {\it irregular coefficients} via weakly interacting particle systems. More precisely, propagation of chaos and convergence rate of Euler-Maruyama scheme associated with the consequent weakly interacting particle systems are investigated for Mckean-Vlasov SDEs, where (i) the diffusion terms are Hölder continuous by taking advantage of Yamada-Watanabe's approximation approach and (ii) the drifts are Hölder continuous by freezing distributions followed by invoking Zvonkin's transformation trick.
Under a Lipschitz condition on distribution dependent coefficients, the central limit theorem and the moderate deviation principle are obtained for solutions of McKean-Vlasov type stochastic differential equations, which extend from the corresponding results for classical stochastic differential equations to the distribution dependent setting.
We present two cubature on Wiener space algorithms for the numerical solution of McKean-Vlasov SDEs with smooth scalar interaction. The analysis hinges on sharp gradient to time-inhomogeneous parabolic PDEs bounds. These bounds may be of independent interest. They extend the classical results of Kusuoka \& Stroock. Both algorithms are tested through two numerical examples.
In this article, we develop integration by parts formulae on Wiener space for solutions of SDEs with general McKean-Vlasov interaction and uniformly elliptic coefficients. These integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of Lions. They allows us to prove the existence of a classical solution to a related PDE with irregular terminal condition. We also develop bounds for the derivatives of the density of the solutions of McKean-Vlasov SDEs.
By using the heat kernel parameter expansion with respect to the frozen SDEs, the intrinsic derivative is estimated for the law of Mckean-Vlasov SDEs with respect to the initial distribution. As an application, the total variation distance between the laws of two solutions is bounded by the Wasserstein distance for initial distributions. These extend some recent results proved for distribution-free noise by using the coupling method and Malliavin calculus.
In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, using the decoupled method, some regularity such as relative entropy and Sobolev's estimate of invariant probability measure are proved. Finally, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained.
The work concerns a type of backward multivalued McKean-Vlasov stochastic differential equations. First, we prove the existence and uniqueness of solutions for backward multivalued McKean-Vlasov stochastic differential equations. Then, it is presented that their solutions depend continuously on the terminal values. Finally, we give a probabilistic interpretation for viscosity solutions of nonlocal quasi-linear parabolic variational inequalities.
We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results th…
Under Lyapunov and monotone conditions, the exponential ergodicity in the induced Wasserstein quasi-distance is proved for a class of fully non-dissipative McKean-Vlasov SDEs, which strengthen some recent results established under dissipative conditions in long distance. Moreover, when the SDE is order-preserving, the exponential ergodicity is derived in the Wasserstein distance induced by one-dimensional increasing functions chosen according to the coefficients of the equation.
The well-posedness and regularity estimates in initial distributions are derived for singular McKean-Vlasov SDEs, where the drift contains a locally standard integrable term and a superlinear term in the spatial variable, and is Lipchitz continuous in the distribution variable with respect to a weighted variation distance. When the superlinear term is strengthened to be Lipschitz continuous, Wangs Harnack inequality is established. These results are new also for the classical Ito SDEs where the coefficients are distribution independent.
We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works, we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some results in the literature on invariant probability measures for functional SDEs without dependence on the law of the solution.
We study a large class of McKean-Vlasov SDEs with drift and diffusion coefficient depending on the density of the solution's time marginal laws in a Nemytskii-type of way. A McKean-Vlasov SDE of this kind arises from the study of the associated nonlinear FPKE, for which is known that there exists a bounded Sobolev-regular Schwartz-distributional solution u. Via the superposition principle, it is already known that there exists a weak solution to the McKean-Vlasov SDE with time marginal densities u. We show that there exists a strong solution the McKean-Vlasov SDE, which is unique among weak so…
We prove the existence of weak solutions to McKean-Vlasov SDEs defined on a domain $D \subseteq \mathbb{R}^d$ with continuous and unbounded coefficients that satisfy Lyapunov type conditions, where the Lyapunov function may depend on measure. We propose a new type of {\em integrated} Lyapunov condition, where the inequality is only required to hold when integrated against the measure on which the Lyapunov function depends , and we show that this is sufficient for the existence of weak solutions to McKean-Vlasov SDEs defined on $D$. The main tool used in the proofs is the concept of a measure d…
Let P2(Rd) be the space of probability measures on Rd with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space Rd*P2(Rd) equipped with the usual derivative in space variable and Lions derivative in distribution. These PDEs are solved by using probabilis- tic arguments developed from [2]. In particular, the path independence of the Girsanov transformation killing the drift term is identified with a nonlinear PDE on Rd*P2(Rd), which includes corresponding results derived earlier for the classical SDEs as special …
Existence and uniqueness are proved for Mckean-Vlasov type distribution dependent SDEs with singular drifts satisfying an integrability condition in space variable and the Lipschitz condition in distribution variable with respect to $W_0$ or $W_0+W_θ$ for some $θ\ge 1$, where $W_0$ is the total variation distance and $W_θ$ is the $L^θ$-Wasserstein distance. This improves some existing results where the drift is either locally bounded in the space variable or continuous in the distribution variable with respect to the Wasserstein distance.
By using Zvonkin's transformation and a two-step fixed point argument in distributions, the well-posedness and regularity estimates are derived for singular McKean-Vlasov SDEs with distribution dependent noise, where the drift contains a term growing linearly in space and distribution and a locally integrable term independent of distribution, while the noise coefficient is weakly differentiable in space and Lipschitz continuous in distribution with respect to the sum of Wasserstein and weighted variation distances. The main results extend existing ones derived for noise coefficients either ind…
We consider a special class of mean field SDEs with common noise which depend on the image of the solution (i.e. the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games, the Feynman-Kac formula is established to solve Schrördinegr type PDEs on $\scr P_2$, and the ergodicity is proved for a class of measure-valued diffusion processes.
We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.
This paper is devoted to deriving the Onsager-Machlup action functional for Mckean-Vlasov stochastic differential equations in a class of norms that dominate $L^2([0,1], \mathbb{R}^d)$, such as supremum norm $\|\cdot\|_{\infty}$, H$\mathrm{\ddot{o}}$lder norms $\|\cdot\|_α$ with $α<\frac{1}{4}$ and $L^p$-norms with $p>4$ are included. Moreover, the corresponding Euler-Lagrange equation for Onsager-Machlup action functional is derived and a example is given.
In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(1/N + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.
In this paper, we are interested in least squares estimator for a class of path-dependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown pa- rameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of our paper lie in three aspects: (i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no long…
We consider a class of Lévy-driven stochastic differential equations (SDEs) with McKean-Vlasov (MK-V) interaction in the drift coefficient. It is assumed that the coefficient is bounded, affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration scheme that app…
In this paper, we aim to study the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we prove the weak convergence of slow process $X^\varepsilon$ in $C([0,T];\mathbb{R}^n)$ towards the limiting process $X$ that is the solution of a distribution dependent stochastic differential equation in which some new drift and diffusion terms compared to the original equation appear. The main contribution is to use two different methods to explicitly characterize the limiting equations respectively. The obtained diffusion coefficients in the limiting …
In this paper we consider a class of {\it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called {\it conditional mean-field SDEs} (CMFSDE) studied previously by the authors [1, 14], but with some fundamental differences. In particular, due to the lack of compactness of the iterated conditional laws, the existing arguments of Schauder's fixed point theorem do not seem to apply in this situation, and the heavy nonlinearity on the conditional laws caused by …
