Mean-field Stochastic Differential Equations Research Articles

Weyl almost periodic solutions in distribution to a mean-field stochastic differential equation driven by fractional Brownian motion

In this paper, we consider a class of mean-field stochastic differential equations driven by both Brownian motion and fractional Brownian motion. Using the Banach fixed point theorem and inequality technique, we obtain sufficient conditions for the existence and uniqueness of Weyl almost periodic solutions in distribution of the equations under consideration.

On Mean Field Stochastic Differential Equations Driven by $$G$$-Brownian Motion with Averaging Principle

On Mean Field Stochastic Differential Equations Driven by $$G$$-Brownian Motion with Averaging Principle

Existence and Uniqueness Theorem for a Class of Mean-Field Stochastic Differential Equations

Existence and Uniqueness Theorem for a Class of Mean-Field Stochastic Differential Equations

Indefinite linear-quadratic optimal control of mean-field stochastic differential equation with jump diffusion: an equivalent cost functional method

Indefinite linear-quadratic optimal control of mean-field stochastic differential equation with jump diffusion: an equivalent cost functional method

Linear-quadratic optimal control problem for mean-field stochastic differential equations with a type of random coefficients

Linear-quadratic optimal control problem for mean-field stochastic differential equations with a type of random coefficients

Relationship between Nash Equilibrium Strategies and H2/H∞ Control of Mean-Field Stochastic Differential Equations with Multiplicative Noise

The relationship between finite-horizon mean-field stochastic H2/H∞ control and Nash equilibrium strategies is investigated in this technical note. First, the finite-horizon mean-field stochastic bounded real lemma (SBRL) is established, which is key to developing the H∞ theory. Second, for mean-field stochastic differential equations (MF-SDEs) with control- and state-dependent noises, it is revealed that the existence of Nash equilibrium strategies is equivalent to the solvability of generalized differential Riccati equations (GDREs). Furthermore, the existence of Nash equilibrium strategies is equivalent to the solvability of H2/H∞ control for MF-SDEs with control- and state-dependent noises. However, for mean-field stochastic systems with disturbance-dependent noises, these two problems are not equivalent. Finally, a sufficient and necessary condition is presented via coupled matrix-valued equations for the finite-horizon H2/H∞ control of mean-field stochastic differential equations with disturbance-dependent noises.

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

Linear-Quadratic Non-Zero Sum Backward Stochastic Differential Game With Overlapping Information

This article studies a linear quadratic non-zero sum stochastic differential game with overlapping information, where the state dynamics are described by a backward stochastic differential equation and the information obtained by two players has a common part but no inclusion relation. The open-loop Nash equilibrium strategy is given by some conditional mean-field stochastic differential equations. In addition, coupled Riccati equations are introduced to express the state feedback form of the Nash equilibrium strategy.

Mean-field stochastic differential equations with a discontinuous diffusion coefficient

Mean-field stochastic differential equations with a discontinuous diffusion coefficient

Mean-field stochastic differential equations driven by <inline-formula><tex-math id="M1">$ G $</tex-math></inline-formula>-Brownian motion

The main purpose of this paper is to analyze the mean-field stochastic differential equations driven by $ G $-Brownian motion (mean-field $ G $-SDEs). We first investigate the existence and uniqueness of the solution for the mean-field $ G $-SDEs by utilizing the Banach fixed point theorem. Moreover, by the method of coupling by change of measures, the Harnack and log-Harnack inequalities for mean-field $ G $-SDEs with additive noise are established.

Periodic Solutions in Distribution of Mean-Field Stochastic Differential Equations

Periodic Solutions in Distribution of Mean-Field Stochastic Differential Equations

Necessary and sufficient conditions in optimal control of mean-field stochastic differential equations with infinite horizon

Abstract We consider an infinite horizon optimal control of a system where the dynamics evolve according to a mean-field stochastic differential equation and the cost functional is also of mean-field type. These are systems where the coefficients depend not only on the state variable, but also on its marginal distribution via some linear functional. Under some concavity assumptions on the coefficients as well as on the Hamiltonian, we are able to prove a verification theorem, which gives a sufficient condition for optimality for a given admissible control. In the absence of concavity, we prove a necessary condition for optimality in the form of a weak Pontryagin maximum principle, given in terms of stationarity of the Hamiltonian.

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process

On the stability of mean-field stochastic differential equations with irregular expectation functional

In this paper, we consider multidimensional mean-field stochastic differential equations where the coefficients depend on the law in the form of a Lebesgue integral with respect to the measure of the solution. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application for optimal control of diffusions. Namely, we propose a result on the approximation of the solution associated to a relaxed control.

A maximum principle for mean-field stochastic control system with noisy observation

This paper is concerned with an optimal control problem driven by mean-field stochastic differential equation, where the state is partially observed via a noisy process. A new feature of the control problem is that the drift coefficient of observation equation is allowed to be a linear growth function, which immediately causes a difficulty in studying the control problem. A backward separation method with a dominated growth rate idea is used to overcome the difficulty. A maximum principle for optimality of the control problem is derived by an approximate technique. An asset–liability management problem for an insurance firm is studied. An optimal premium strategy in a feedback form is explicitly obtained.

Dynamic optimization problems for mean-field stochastic large-population systems

This paper considers dynamic optimization problems for a class of control average meanfield stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation with individual noise and common noise, and the weight coefficients in the corresponding cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation. Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence equation systems are discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.

Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

Numerical methods for mean-field stochastic differential equations with jumps

In this paper, we are devoted to the numerical methods for mean-field stochastic differential equations with jumps (MSDEJs). By combining with the mean-field Itô formula (see Sun, Yang, and Zhao, Numer. Math. Theor. Meth. Appl., 10, pp. 798–828 (2017)), we first develop the Itô formula and further construct the Itô-Taylor expansion for MSDEJs. Then based on the Itô-Taylor expansions, we propose the strong order γ and the weak order η Itô-Taylor schemes for MSDEJs. We theoretically prove the strong convergence rate γ of the strong order γ Itô-Taylor scheme and the weak convergence rate η of the weak order η Itô-Taylor scheme, respectively. Some numerical tests are also presented to verify our theoretical conclusions.

Partial derivative with respect to the measure and its application to general controlled mean-field systems

Let (E,E) be an arbitrary measurable space. The paper first focuses on studying the partial derivative of a function f:P2,0(Rd×E)→R defined on the space of probability measures μ over (Rd×E,B(Rd)⊗E) whose first marginal μ1≔μ(⋅×E) has a finite second order moment. This partial derivative is taken with respect to q(dx,z), where μ has the disintegration μ(dxdz)=q(dx,z)μ2(dz) with respect to its second marginal μ2(⋅)=μ(Rd×⋅). Simplifying the language, we will speak of the derivative with respect to the law μ conditioned to its second marginal. Our results extend those of the derivative of a function g:P2(Rd)→R over the space of probability measures with finite second order moment by P.L. Lions (see Lions (2013)) but cover also as a particular case recent approaches considering E=Rk and supposing the differentiability of f over P2(Rd×Rk), in order to use the derivative ∂μf to define the partial derivative (∂μf)1. The second part of the paper focuses on investigating a stochastic maximum principle, where the controlled state process is driven by a general mean-field stochastic differential equation with partial information. The control set is just supposed to be a measurable space, and the coefficients of the controlled system, i.e., those of the dynamics as well as of the cost functional, depend on the controlled state process X, the control v, a partial information on X, as well as on the joint law of (X,v). Through considering a new second-order variational equation and the corresponding second-order adjoint equation, and a totally new method to prove the estimate for the solution of the first-order variational equation, the optimal principle is proved through spike variation of an optimal control and with the help of the tailor-made form of second-order expansion. We emphasize that in our assumptions we do not need any regularity of the coefficients neither in the control variable nor with respect to the law of the control process.

A Mean-Field Linear-Quadratic Stochastic Stackelberg Differential Game with one Leader and Two Followers

This paper is concerned with a linear-quadratic (LQ) stochastic Stackelberg differential game with one leader and two followers, where the game system is governed by a mean-field stochastic differential equation (MF-SDE). By maximum principle and verification theorem, the open-loop Stackelberg solution is expressed as a feedback form of the state and its mean with the help of three systems of Riccati equations.