McKean Vlasov Stochastic Differential Equations Research Articles

Multilevel importance sampling for rare events associated with the McKean–Vlasov equation

AbstractThis work combines multilevel Monte Carlo with importance sampling to estimate rare-event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to a broad class of McKean–Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator introduced in this context in Ben Rached et al. (Stat Comput, 2024. https://doi.org/10.1007/s11222-024-10497-3) to the multilevel setting. We formulate a novel multilevel DLMC estimator and perform a comprehensive cost-error analysis yielding new and improved complexity results. Crucially, we devise an antithetic sampler to estimate level differences guaranteeing reduced computational complexity for the multilevel DLMC estimator compared with the single-level DLMC estimator. To address rare events, we apply the importance sampling scheme, obtained via stochastic optimal control in Ben Rached et al. (2024), over all levels of the multilevel DLMC estimator. Combining importance sampling and multilevel DLMC reduces computational complexity by one order and drastically reduces the associated constant compared to the single-level DLMC estimator without importance sampling. We illustrate the effectiveness of the proposed multilevel DLMC estimator on the Kuramoto model from statistical physics with Lipschitz observables, confirming the reduced complexity from $${\mathcal {O}(\textrm{TOL}_{\textrm{r}}^{-4})}$$ O ( TOL r - 4 ) for the single-level DLMC estimator to $${\mathcal {O}(\textrm{TOL}_{\textrm{r}}^{-3})}$$ O ( TOL r - 3 ) while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $$\textrm{TOL}_{\textrm{r}}$$ TOL r .

The Onsager-Machlup Action Functional for Degenerate McKean-Vlasov Stochastic Differential Equations

The purpose of this paper is to investigate the existence of the Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations. To this end, we first derive Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations with constant diffusion in a broad set of norms by Girsanov transformation, some conditioned exponential inequalities and It $\mathrm{\hat{o}}$ formulas for distribution dependent functional. Then an example is given to illustrate our results.

Large deviation for slow-fast McKean-Vlasov stochastic differential equations driven by fractional Brownian motions and Brownian motions

Large deviation for slow-fast McKean-Vlasov stochastic differential equations driven by fractional Brownian motions and Brownian motions

On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients

This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by Lévy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of Lévy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.

Stabilisation of Mckean–Vlasov stochastic differential equations by aperiodically intermittent control

This paper mainly studies the mean-square exponential stabilisation for McKean–Vlasov stochastic differential equations (MVSDEs, in short) by aperiodically intermittent control, which includes periodically intermittent control as a special case. Sufficient conditions are given to make the unstable MVSDEs be exponentially stable. As an application, the stabilisation criterions for the mean-field Cohen-Grossberg neural network model via aperiodically intermittent control are established.

Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document} with explicit Lyapunov exponents.

Parametric inference for ergodic McKean-Vlasov stochastic differential equations

Parametric inference for ergodic McKean-Vlasov stochastic differential equations

A necessary conditions for optimal singular control of McKean-Vlasov stochastic differential equations driven by spatial parameters local martingale

In this paper, we focus on stochastic singular control problems involving McKean-Vlasov stochastic differential equations driven by a spatially parameterized continuous local martingale. The drift coefficient in these equations depends on the state of the solution process and its law. The control variable consists of two components: an absolutely continuous control and a singular one. Firstly, under Lipschitz conditions, we establish the existence and uniqueness of its strong solution. Next, we derive the necessary conditions for optimal singular control under the assumption that the control domain is convex. These optimality conditions differ from the classical ones in the sense that here the adjoint equation is a McKean-Vlasov backward stochastic differential equation driven by a continuous local martingale with spatial parameters. The proof of our result is based on the derivatives of the local martingale with respect to spatial parameters and the derivative of the drift coefficient with respect to the probability measure.

Rectified deep neural networks overcome the curse of dimensionality when approximating solutions of McKean–Vlasov stochastic differential equations

In this paper we prove that rectified deep neural networks do not suffer from the curse of dimensionality when approximating McKean–Vlasov SDEs in the sense that the number of parameters in the deep neural networks only grows polynomially in the space dimension d of the SDE and the reciprocal of the accuracy ϵ.

Optimal Estimation of a Signal Generated Using a Dynamical System Modeled with McKean-Vlasov Stochastic Differential Equations.

We consider, in this paper, the problem of state estimation for a class of dynamical systems governed via continuous-time McKean-Vlasov stochastic differential equations. The estimation problem is stated and solved under an H2 norm setting. We adopt a Riccati-based approach in order to solve the optimal estimation problem.

Strong convergence of Euler–Maruyama schemes for doubly perturbed McKean–Vlasov stochastic differential equations

In this paper, we develop strong convergence of the Euler–Maruyama (EM) scheme for approximating the doubly perturbed McKean–Vlasov stochastic differential equations. In contrast to the existing work, a novel feature is that we use more general conditions for parameters α and β. To obtain the desired approximation, this paper also proves the existence and uniqueness of strong solution for this class of McKean–Vlasov SDEs. Combining with the results of propagation of chaos, the overall convergence rate is obtained for the EM scheme. Finally, two numerical examples are provided to demonstrate our results.

Impulse Control of Conditional McKean–Vlasov Jump Diffusions

In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.

Weak convergence of Mckean-Vlasov stochastic differential equations with two-time-scale Markov switching

Weak convergence of Mckean-Vlasov stochastic differential equations with two-time-scale Markov switching

Large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations

In this paper, we study large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations. First of all, we establish the large deviation principle for the space-distribution dependent Zakai equation by a weak convergence approach. Then based on the obtained result and the relationship between the space-distribution dependent Zakai equation and the space-distribution dependent Kushner-Stratonovich equation, the large deviation principle for the latter is proved.

Stochastic averaging principle for McKean–Vlasov SDEs driven by Lévy noise

In this paper, we study McKean–Vlasov stochastic differential equations driven by Lévy processes. Firstly, under the non-Lipschitz condition which include classical Lipschitz conditions as special cases, we establish the existence and uniqueness for solutions of McKean–Vlasov stochastic differential equations using Carathéodory approximation. Then under certain averaging conditions, we establish a stochastic averaging principle for McKean–Vlasov stochastic differential equations driven by Lévy processes. We find that the solutions to stochastic systems concerned with Lévy noise can be approximated by solutions to averaged McKean–Vlasov stochastic differential equations driven by Lévy processes in the sense of convergence in [Formula: see text]th moment.

The locally homeomorphic property of McKean-Vlasov SDEs under the global Lipschitz condition

In this paper, we establish the locally diffeomorphic property of the solution to McKean-Vlasov stochastic differential equations defined in the Euclidean space. Our approach is built upon the insightful ideas put forth by Kunita. We observe that although the coefficients satisfy the global Lipschitz condition and some suitable regularity condition, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time, which sets McKean-Vlasov stochastic differential equations apart significantly from classical stochastic differential equations. Finally, we provide an example to complement our results.

Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean–Vlasov equations and interacting particle systems

Abstract In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean–Vlasov stochastic differential equations, possibly with a drift term exhibiting super-linear growth in the state component. We prove strong convergence of order one and moment stability, making use of techniques from variational calculus on the space of probability measures with finite second-order moments. Then, we introduce an antithetic multilevel Milstein scheme, which leads to optimal complexity estimators for expected functionals of solutions to delay McKean–Vlasov equations without the need to simulate Lévy areas.

The mean‐field linear quadratic optimal control problem for stochastic systems controlled by impulses

AbstractIn the present note, we state and solve the linear quadratic (LQ) control problem for a class of McKean–Vlasov stochastic differential equations for which the control actions are of impulsive nature. After reformulating the original optimization problem using an orthogonal decomposition of the state variables, we introduce an adequately defined system of two coupled matrix Lyapunov‐type differential equations with jumps. Such equations are central for the definition of the optimal feedback gains corresponding to the closed‐loop representation of the optimal LQ control.

Limit theorems of invariant measures for multivalued McKean-Vlasov stochastic differential equations

The work concerns invariant measures for multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the exponential ergodicity of these equations. Then for a sequence of these equations, when their coefficients converge in the suitable sense, the strong convergence of corresponding strong solutions are presented. Finally, based on the convergence of these solutions, we establish the convergence of corresponding invariant measures with respect to the 1-Wasserstein distance.

Ergodic control of McKean–Vlasov SDEs and associated Bellman equation

We consider the ergodic control problem for McKean-Vlasov stochastic differential equations and prove the existence and uniqueness of the viscosity solution to the associated fully nonlinear HJB equation in a lifted sense. Furthermore, as the time horizon goes to infinity, we show that the solutions of finite-horizon time-averaging optimal control problems converge to that of the ergodic control problem. Our results require dissipativity conditions and dissipativity-like conditions on distribution variables of both drift and diffusion coefficients.