McKean Vlasov Stochastic Differential Equations Research Articles

Well-posedness and numerical schemes for one-dimensional McKean–Vlasov equations and interacting particle systems with discontinuous drift

In this paper, we first establish well-posedness results for one-dimensional McKean–Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler–Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean–Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order 1/2 known for the Lipschitz case cannot be recovered for all presented schemes.

Large and Moderate Deviation Principles for McKean-Vlasov SDEs with Jumps

In this paper, we consider McKean-Vlasov stochastic differential equations (MVSDEs) driven by Lévy noise. By identifying the right equations satisfied by the solutions of the MVSDEs with shifted driving Lévy noise, we build up a framework to fully apply the weak convergence method to establish large and moderate deviation principles for MVSDEs. In the case of ordinary SDEs, the rate function is calculated by using the solutions of the corresponding skeleton equations simply replacing the noise by the elements of the Cameron-Martin space. It turns out that the correct rate function for MVSDEs is defined through the solutions of skeleton equations replacing the noise by smooth functions and replacing the distributions involved in the equation by the distribution of the solution of the corresponding deterministic equation (without the noise). This is somehow surprising. With this approach, we obtain large and moderate deviation principles for much wider classes of MVSDEs in comparison with the existing literature see Dos Reis et al. (Ann. Appl. Probab. 29, 1487–1540, 2019).

A flexible split‐step scheme for solving McKean‐Vlasov stochastic differential equations

We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation. The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [1] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented, including a comparative analysis to other known algorithms for this class (Taming and Adaptive time-stepping) across parallel and non-parallel implementations.

Parameter Estimation of Path-Dependent McKean-Vlasov Stochastic Differential Equations

This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Finally, we estimate the errors between solutions of these equations and that of their numerical equations.

Strong convergence rates in averaging principle for slow-fast McKean-Vlasov SPDEs

In this paper, we aim to study the asymptotic behavior for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic p-Laplace type equation and also some McKean-Vlasov stochastic differential equations.

Strong convergence of Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations under local Lipschitz conditions of state variables

AbstractThis paper develops strong convergence of the Euler–Maruyama (EM) schemes for approximating McKean–Vlasov stochastic differential equations (SDEs). In contrast to the existing work, a novel feature is the use of a much weaker condition—local Lipschitzian in the state variable, but under uniform linear growth assumption. To obtain the desired approximation, the paper first establishes the existence and uniqueness of solutions of the original McKean–Vlasov SDE using a Euler-like sequence of interpolations and partition of the sample space. Then, the paper returns to the analysis of the EM scheme for approximating solutions of McKean–Vlasov SDEs. A strong convergence theorem is established. Moreover, the convergence rates under global conditions are obtained.

Least squares estimation for delay McKean–Vlasov stochastic differential equations and interacting particle systems

Least squares estimation for delay McKean–Vlasov stochastic differential equations and interacting particle systems

McKean\u2013Vlasov type stochastic differential equations arising from the random vortex method

We study a class of McKean–Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean–Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot–Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics.

Multilevel Picard approximations for McKean-Vlasov stochastic differential equations

In the literature there exist approximation methods for McKean-Vlasov stochastic differential equations which have a computational effort of order 3. In this article we introduce full-history recursive multilevel Picard approximations for McKean-Vlasov stochastic differential equations. We prove that these MLP approximations have computational effort of order 2+ which is essentially optimal in high dimensions.

Necessary conditions for partially observed optimal control of general McKean–Vlasov stochastic differential equations with jumps

In this paper, we establish necessary conditions of optimality for partially observed optimal control problems of Mckean–Vlasov type. The system is described by a controlled stochastic differential equation governed by Poisson random measure and an independent Brownian motion. The coefficients of the McKean–Vlasov system depend on the state of the solution process as well as of its probability law and the control variable. The proof of our result is based on Girsanov's theorem, variational equations and derivatives with respect to probability measure under convexity assumption. At the end of this paper, we apply our stochastic maximum principle to study partially observed linear quadratic control problem of McKean–Vlasov type with jumps and derive the explicit expression of the optimal control.

A higher order weak approximation of McKean–Vlasov type SDEs

The paper introduces a new weak approximation algorithm for stochastic differential equations (SDEs) of McKean–Vlasov type. The arbitrary order discretization scheme is available and is given using Malliavin weights, certain polynomial weights of Brownian motion, which play a role as correction of the approximation. The new weak approximation scheme works even if the test function is not smooth. In other words, the expectation of irregular functionals of McKean–Vlasov SDEs such as probability distribution functions are approximated through the proposed scheme. The effectiveness of the higher order scheme is confirmed by numerical examples for McKean–Vlasov SDEs including the Kuramoto model.

An adaptive Euler–Maruyama scheme for McKean–Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh–Nagumo model

In this paper, we introduce fully implementable, adaptive Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component. We prove moment stability of the discretised processes and a strong convergence rate of 1/2. Several numerical examples, centred around a mean-field model for FitzHugh–Nagumo neurons, illustrate that the standard uniform scheme fails and that the adaptive approach shows in most cases superior performance to tamed approximation schemes. In addition, we introduce and analyse an adaptive Milstein scheme for a certain sub-class of McKean–Vlasov SDEs with linear measure-dependence of the drift.

Information upper bound for McKean-Vlasov stochastic differential equations.

We develop an information-theoretic framework to quantify information upper bound for the probability distributions of the solutions to the McKean-Vlasov stochastic differential equations. More precisely, we derive the information upper bound in terms of Kullback-Leibler divergence, which characterizes the entropy of the probability distributions of the solutions to McKean-Vlasov stochastic differential equations relative to the joint distributions of mean-field particle systems. The order of information upper bound is also figured out.

Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps

In this paper, the path independent property of additive functionals of McKean–Vlasov stochastic differential equations with jumps is characterized by nonlinear partial integro-differential equations involving [Formula: see text]-derivatives with respect to probability measures introduced by Lions. Our result extends the recent work16 by Ren and Wang where their concerned McKean–Vlasov stochastic differential equations are driven by Brownian motions.

Approximations of McKean–Vlasov Stochastic Differential Equations with Irregular Coefficients

The goal of this paper is to approximate two kinds of McKean–Vlasov stochastic differential equations (SDEs) with 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 (1) the diffusion terms are Holder continuous by taking advantage of Yamada–Watanabe’s approximation approach and (2) the drifts are Holder continuous by freezing distributions followed by invoking Zvonkin’s transformation trick.

Strong convergence order for slow–fast McKean–Vlasov stochastic differential equations

In this paper, we consider the averaging principle for a class of McKean–Vlasov stochastic differential equations with slow and fast time-scales. Under some proper assumptions on the coefficients, we first prove that the slow component strongly converges to the solution of the corresponding averaged equation with convergence order 1/3 using the approach of time discretization. Furthermore, under stronger regularity conditions on the coefficients, we use the technique of Poisson equation to improve the order to 1/2, which is the optimal order of strong convergence in general.

Simulation of McKean–Vlasov SDEs with super-linear growth

Abstract We present two fully probabilistic Euler schemes, one explicit and one implicit, for the simulation of McKean–Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth and random initial condition. We provide a pathwise propagation of chaos result and show strong convergence for both schemes on the consequent particle system. The explicit scheme attains the standard $1/2$ rate in stepsize. From a technical point of view, we successfully use stopping times to prove the convergence of the implicit method; although we avoid them altogether for the explicit one. The combination of particle interactions and random initial condition makes the proofs technically more involved. Numerical tests recover the theoretical convergence rates and illustrate a computational complexity advantage of the explicit over the implicit scheme. Comparative analysis is carried out on a stylized non-Lipschitz MV-SDE and a mean-field model for FitzHugh–Nagumo neurons. We provide numerical tests illustrating particle corruption effect where one single particle diverging can ‘corrupt’ the whole particle system. Moreover, the more particles in the system the more likely this divergence is to occur.

Stability and prevalence of McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients

Abstract We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.

On explicit Milstein-type scheme for McKean–Vlasov stochastic differential equations with super-linear drift coefficient

We develop an explicit Milstein-type scheme for McKean-Vlasov stochastic differential equations using the notion of derivative with respect to measure introduced by Lions and discussed in \cite{cardaliaguet2013}. The drift coefficient is allowed to grow super-linearly in the space variable. Further, both drift and diffusion coefficients are assumed to be only once differentiable in variables corresponding to space and measure. The rate of strong convergence is shown to be equal to $1.0$ without using It\^o's formula for functions depending on measure. The challenges arising due to the dependence of coefficients on measure are tackled and our findings are consistent with the analogous results for stochastic differential equations.

First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

In this paper, we derive fully implementable first-order time-stepping schemes for McKean-Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean-Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.