Mean-field Forward-backward Stochastic Differential Equations Research Articles

Properties of solution for fully coupled fractional mean-field forward-backward stochastic differential equation

Properties of solution for fully coupled fractional mean-field forward-backward stochastic differential equation

Partially observed mean-field Stackelberg stochastic differential game with two followers

This paper studies a partially observed stochastic Stackelberg game with two followers, where state satisfies a linear stochastic differential equation of mean-field type, and cost functionals are quadratic. Using decomposition technique and backward separation approach, we derive the followers' optimal strategy. The leader focuses on an optimal control problem driven by a fully coupled mean-field forward–backward stochastic differential equation with conditional expectation. By layer-by-layer decomposition technique, some inequalities and Riccati equations, we not only get the existence and uniqueness of solution to the corresponding Hamiltonian system but also give a feedback form of Stackelberg solution. Finally, we tackle a government debt problem by the above theoretical results.

Fully-coupled mean-field FBSDE and maximum principle for related optimal control problem

In this paper, motivated by the study of Stackelberg differential game of McKean–Vlasov type, we first investigate the solvability for the mean-field forward–backward stochastic differential equation (MF-FBSDE), which involves one McKean–Vlasov stochastic differential equation (SDE) and one mean-field type backward stochastic differential equation (BSDE). Under some weakly coupled conditions, we use a purely probabilistic approach to prove the well-posedness of such MF-FBSDE with arbitrary fixed time horizon. Then we consider a mean-field control problem, where the state dynamic is given by controlled MF-FBSDE. We establish the stochastic maximum principle and verification theorem for this optimal control problem. We mention that our weakly coupled conditions introduced in studying the solvability for the MF-FBSDE can be applied to simultaneously guarantee the well-posedness of variational equation and adjoint equation which are both fully-coupled MF-FBSDE. As an application, we consider a Stackelberg differential game whose system is described by a McKean–Vlasov SDE. We give a necessary condition for the existence of Stackelberg solutions.

Linear-Quadratic-Gaussian mean-field controls of social optima

<p style='text-indent:20px;'>This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be <i>indefinite</i> in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some <i>fully-coupled</i> variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with <i>embedding representation</i>. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.</p>

Mixed Social Optima and Nash Equilibrium in Linear-Quadratic-Gaussian Mean-Field System

This article investigates a class of mixed stochastic linear-quadratic-Gaussian social optimization and Nash game in the context of a large-scale system. Two types of interactive agents are involved: a major agent and a large number of weakly coupled minor agents. All minor agents are cooperative to minimize the social cost as the sum of their individual costs, whereas such social cost is conflictive to that of the major agent. Thus, the major agent and all minor agents are further competitive to reach some nonzero-sum Nash equilibrium. Applying the mean-field approximations and person-by-person optimality, we obtain auxiliary control problems for the major agent and minor agents, respectively. The decentralized social strategy is derived by a class of new consistency condition (CC) system, which consists of mean-field forward–backward stochastic differential equations. The well-posedness of CC system is obtained by the discounting method. The related asymptotic social optimality for minor agents and Nash equilibrium for major–minor agents are also verified.

Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations

We study a coupled fractional mean-field forward–backward stochastic differential equation (MF-FBSDE), in which the coefficients involved could also depend upon the distribution of the solution (X, Y), and which contains a special structure η. We prove the existence and uniqueness of a solution of the fractional MF-FBSDE by using the method of continuation.

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.

Mean-field linear-quadratic stochastic differential games in an infinite horizon

This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

Pareto-Optimal Strategy for Linear Mean-Field Stochastic Systems With H∞ Constraint.

This article presents results on designing the Pareto-optimal strategy under H∞ constraint for the linear mean-field stochastic systems disturbed by external disturbances. First, combining the stochastic H∞ control theory with the stochastic mean-field theory, we derive the stochastic bounded real lemma (SBRL) of our considered linear mean-field stochastic systems with the stochastic initial condition. Second, we use the mean-field forward-backward stochastic differential equation to solve the mean-field linear quadratic Pareto-optimal problem with indefinite cost functionals. It is proved that the existence of a closed-loop Pareto-optimal strategy is equivalent to the solvability of the coupled generalized differential Riccati equations when some conditions are satisfied. Finally, a necessary and sufficient condition for the Pareto-optimal strategy under the H∞ constraint is researched by four-coupled matrix-valued equations. Besides, we also obtain the Pareto frontier for the mean-field stochastic system with only state-dependent noise. A practical example is presented to show the effectiveness of our main results.

Exact controllability of linear mean‐field stochastic systems and observability inequality for mean‐field backward stochastic differential equations

AbstractThis paper is concerned with the exact controllability of linear mean‐field stochastic systems with time‐variant random coefficients. We prove that the exact controllability, the validity of the observability inequality for the dual equation, the unique solvability of a family of optimal control problems, the unique solvability of a family of mean‐field forward‐backward stochastic differential equations (MF‐FBSDEs), and the unique solvability of a family of norm optimal control problems are all equivalent. Therefore, some approaches are provided to investigate the issue of exact controllability.

A partial information linear-quadratic optimal control problem of backward stochastic differential equation with its applications

In this paper, we investigate a kind of partial information linear-quadratic optimal control problem driven by a backward stochastic differential equation, where the state equation and the cost functional contain diffusion terms. Using maximum principle, we derive the corresponding Hamiltonian system, which is a conditional mean-field forward-backward stochastic differential equation. By the backward separation approach and the filtering technique, we get two Riccati equations, and a backward and a forward optimal filtering equations. Then a feedback form of optimal control is obtained.We also extend thecontrol problem to the case of mean-field backward stochastic differential equation under partial information.A corresponding feedback form of optimal control is also obtained.

Maximum Principle for Near-Optimality of Mean-Field FBSDEs

The present paper concerns with a near-optimal control problem for systems governed by mean-field forward-backward stochastic differential equations (FBSDEs) with mixed initial-terminal conditions. Utilizing Ekeland’s variational principle as well as the reduction method, the necessary and sufficient near-optimality conditions are established in the form of Pontryagin’s type. The results are obtained under restriction on the convexity of the control domain. As an application, a linear-quadratic stochastic control problem is solved explicitly.

A Tale of a Principal and Many, Many Agents

In this paper, we investigate a moral hazard problem in finite time with lump-sum and continuous payments, involving infinitely many agents with mean-field type interactions, hired by one principal. By reinterpreting the mean-field game faced by each agent in terms of a mean-field forward-backward stochastic differential equation (FBSDE), we are able to rewrite the principal’s problem as a control problem of the McKean-Vlasov stochastic differential equations. We review one general approach to tackling it, introduced recently using dynamic programming and Hamilton-Jacobi-Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle. We solve completely and explicitly the problem in special cases, going beyond the usual linear-quadratic framework. We finally show in our examples that the optimal contract in the N-players’ model converges to the mean-field optimal contract when the number of agents goes to +∞.

Linear quadratic mean field game with control input constraint

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subsetΓof full space ℝm. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators onΓ. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The relatedϵ-Nash equilibrium property is also verified.

Asymptotic analysis of mean field games with small common noise

In this paper, we consider a mean field game (MFG) model perturbed by small common noise. Our goal is to give an approximation of the Nash equilibrium strategy of this game using a solution from the original no common noise MFG whose solution can be obtained through a coupled system of partial differential equations. We characterize the first order approximation via linear mean-field forward-backward stochastic differential equations whose solution is a centered Gaussian process with respect to the common noise. The first order approximate strategy can be described as follows: at time $t \in [0,T]$, applying the original MFG optimal strategy for a sub game over $[t,T]$ with the initial being the current state and distribution. We then show that this strategy gives an approximate Nash equilibrium of order $\epsilon^2$.

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents with mean-field interactions. The individual admissible controls are constrained in closed convex subsets $Γ k$ of full space $R m$. The decentralized strategies for individual agents and consistency condition system are represented in an unified manner via a class of mean-field forward-backward stochastic differential equations involving projection operators on $Γ k$. The well-posedness of consistency system is established in the local and global cases both by the contraction mapping and discounting method respectively. The related $e$−Nash equilibrium property is also verified.

An optimal control problem for mean-field forward–backward stochastic differential equation with noisy observation

This article is concerned with an optimal control problem derived by mean-field forward–backward stochastic differential equation with noisy observation, where the drift coefficients of the state equation and the observation equation are linear with respect to the state and its expectation. The control problem is different from the existing literature concerning optimal control for mean-field stochastic systems, and has more applications in mathematical finance, e.g., asset–liability management problem with recursive utility, systematic risk model. Using a backward separation method with a decomposition technique, two optimality conditions along with two coupled forward–backward optimal filters are derived. Linear–quadratic optimal control problems for mean-field forward–backward stochastic differential equations are studied.

Variational principle for stochastic singular control of mean-field Lévy-forward-backward system driven by orthogonal Teugels martingales with application

We consider stochastic singular control for mean-field forward-backward stochastic differential equations, driven by orthogonal Teugels martingales associated with some Levy processes having moments of all orders and an independent Brownian motion. Under partial information, necessary and sufficient conditions for optimality in the form of maximum principle for this mean-field system are established by means of convex variation methods and duality techniques. As an illustration, this paper studies a partial information mean-variance portfolio selection problem driven by orthogonal Teugels martingales associated with gamma process as Levy process of bounded variation.

Variational principle for stochastic singular control of mean-field Lévy-forward-backward system driven by orthogonal Teugels martingales with application

We consider stochastic singular control for mean-field forward-backward stochastic differential equations, driven by orthogonal Teugels martingales associated with some Lévy processes having moments of all orders and an independent Brownian motion. Under partial information, necessary and sufficient conditions for optimality in the form of maximum principle for this mean-field system are established by means of convex variation methods and duality techniques. As an illustration, this paper studies a partial information mean-variance portfolio selection problem driven by orthogonal Teugels martingales associated with gamma process as Lévy process of bounded variation.

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper, we consider an optimal control problem with state constraints, where the control system is described by a mean-field forward-backward stochastic differential equation (MFFBSDE, for short) and the admissible control is mean-field type. Making full use of the backward stochastic differential equation theory, we transform the original control system into an equivalent backward form, i.e., the equations in the control system are all backward. In addition, Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control. We also study a stochastic linear quadratic control problem with state constraints.