This work investigates the optimal control problem for reflected McKean-Vlasov SDEs and the viscosity solutions to Hamilton-Jacobi-Bellman(HJB) equations on the Wasserstein space in terms of intrinsic derivative. It follows from the flow property of reflected McKean-Vlasov SDEs that the dynamic programming principle holds. Applying the decoupling method and the heat kernel estimates for parabolic equations, we show that the value function is a viscosity solution to an appropriate HJB equation on the Wasserstein space, where the characterization of absolutely continuous curves on the Wasserstein space by the continuity equations plays an important role. To establish the uniqueness of viscosity solution, we generalize the construction of a distance like function initiated in Burzoni et al.(SICON, 2020) to the Wasserstein space over multidimensional space and show its effectiveness to cope with HJB equations in terms of intrinsic derivative on the Wasserstein space.
We consider conditional McKean-Vlasov stochastic differential equations (SDEs), such as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs satisfy non-linear stochastic partial differential equations (SPDEs) of the second order, whereas the laws of the conditional time-marginals follow Fokker-Planck equations on the space of probability measures. We prove two superposition principles: The first establishes that any solution of the SPDE can be…
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.
We consider a mean field game with common noise in which the diffusion coefficients may be controlled. We prove existence of a weak relaxed solution under some continuity conditions on the coefficients. We then show that, when there is no common noise, the solution of this mean field game is characterized by a McKean-Vlasov type second order backward SDE.
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.
This paper rigorously connects the problem of optimal control of McKean-Vlasov dynamics with large systems of interacting controlled state processes. Precisely, the empirical distributions of near-optimal control-state pairs for the $n$-state systems, as $n$ tends to infinity, admit limit points in distribution (if the objective functions are suitably coercive), and every such limit is supported on the set of optimal control-state pairs for the McKean-Vlasov problem. Conversely, any distribution on the set of optimal control-state pairs for the McKean-Vlasov problem can be realized as a limit …
This work is concerned with the existence of mild solutions to non-linear Fokker-Planck equations with fractional Laplace operator $(-Δ)^s$ for $s\in\left(\frac12,1\right)$. The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean-Vlasov equations with Lévy-Noise, as well as the Markov property for their laws are proved.
In this paper we study the sensitivity of nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the results obtained for nonlinear PDE of McKean-Vlasov type generated by stable-like processes in the previous works. The motivation for studying sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.
The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate an Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of conditional McKean-Vlasov jump diffusions. Th…
We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean-Vlasov control problem.
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.
We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean--Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution.
In this paper, we show existence and uniqueness of solutions of the infinite horizon McKean-Vlasov FBSDEs using two different methods, which lead to two different sets of assumptions. We use these results to solve the infinite horizon mean field type control problems and mean field games.
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…
In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and Röckner \cite{Kr-Ro} to the case of mixed $L^p$-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel i…
We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of…
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.
The aim of this paper is to introduce several new particle representations for \textit{ergodic} McKean-Vlasov SDEs. We construct new algorithms by leveraging recent progress in weak convergence analysis of interacting particle system. We present detailed analysis of errors and associated costs of various estimators, highlighting key differences between long-time simulations of linear (classical SDEs) versus non-linear (Mckean-Vlasov SDEs) process.
In Talay and Tomasevic [20] we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cutoff. It involved an original type of McKean-Vlasov interaction kernel which involved all the past time marginals of its probability distribution in a singular way. In the present paper, we study this McKean-Vlasov representation in the two-dimensional case. In this setting there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the McKean-Vlasov equation unde…
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.
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non dens…
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 the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.
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.
The Fleming-Viot particle system consists of $N$ identical particles diffusing in a domain $U \subset \mathbb{R}^d$. Whenever a particle hits the boundary $\partial U$, that particle jumps onto another particle in the interior. It is known that this system provides a particle representation for both the Quasi-Stationary Distribution (QSD) and the distribution conditioned on survival for a given diffusion killed at the boundary of its domain. We extend these results to the case of McKean-Vlasov dynamics. We prove that the law conditioned on survival of a given McKean-Vlasov process killed on th…
On Solutions to Graphon McKean-Vlasov SDEs of Nemytskii-type
Bielefeld University（比勒费尔德大学）；Rice University（莱斯大学）；Academy of Mathematics and System Sciences, CAS（中国科学院数学与系统科学研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean-Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for open-loop controls. We also show that the upper and lower v…
In this paper, we present sufficient conditions and criteria to establish the large and moderate deviation principle of multivalued McKean-Vlasov stochastic differential equation by means of the weak convergence method.
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…
