Mean-field Backward Stochastic Differential Equations Research Articles

Existence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with Stochastic Linear Growth Coefficients

We deal with fractional mean field backwardWe deal with fractional mean field backward stochastic differential equations with hurst parameter $H\in (\frac{1}{2},1)$ when the coefficient $f$ satisfy a stochastic Lipschitz conditions, we prove the existence and uniqueness of solution and provide a comparison theorem. Via an approximation and comparison theorem, we show the existence of a minimal solution when the drift satisfies a stochastic growth condition.

Solvability of a class of mean-field BSDEs with quadratic growth

In this paper, we study the multi-dimensional mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. Under small terminal value, the existence and uniqueness are proved for the multi-dimensional situation when the generator f(t,Y,E[Y],Z,E[Z]) is of quadratic growth with respect to the last four items, using some new methods. Besides, a kind of comparison theorem is obtained.

Mean-field backward stochastic differential equations and applications

In this paper we study the linear mean-field backward stochastic differential equations (mean-field BSDE) of the form (0.1) d Y ( t ) = − [ α 1 ( t ) Y ( t ) + β 1 ( t ) Z ( t ) + ∫ R 0 η 1 ( t , ζ ) K ( t , ζ ) ν ( d ζ ) + α 2 ( t ) E [ Y ( t ) ] + β 2 ( t ) E [ Z ( t ) ] + ∫ R 0 η 2 ( t , ζ ) E [ K ( t , ζ ) ] ν ( d ζ ) + γ ( t ) ] d t + Z ( t ) d B ( t ) + ∫ R 0 K ( t , ζ ) N ̃ ( d t , d ζ ) , t ∈ 0 , T , Y ( T ) = ξ . where ( Y , Z , K ) is the unknown solution triplet, B is a Brownian motion, N ̃ is a compensated Poisson random measure, independent of B . We prove the existence and uniqueness of the solution triplet ( Y , Z , K ) of such systems. Then we give an explicit formula for the first component Y ( t ) by using partial Malliavin derivatives. To illustrate our result we apply them to study a mean-field recursive utility optimization problem in finance.

Mean-field backward stochastic differential equations with reflection and related nonlocal PDEs in a convex domain

In this paper, we present the unique solvability of a type of reflected mean-field backward stochastic differential equation (RMFBSDE) in a convex domain. Moreover, employing the techniques of stochastic flow and penalization, we present a probabilistic interpretation in the Sobolev sense for a kind of reflected nonlocal partial differential equation (PDE) in a convex domain. The results of this paper provide the foundation of studying control problems of constrained mean-field systems.

Graphon Mean-Field Backward Stochastic Differential Equations With Jumps and Associated Dynamic Risk Measures

Graphon Mean-Field Backward Stochastic Differential Equations With Jumps and Associated Dynamic Risk Measures

Mean-Field Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

In this paper, we study a new class of equations called mean-field backward stochastic differential equations (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H > 1/2. First, the existence and uniqueness of this class of BSDEs are obtained. Second, a comparison theorem of the solutions is established. Third, as an application, we connect this class of BSDEs with a nonlocal partial differential equation (PDE, for short), and derive a relationship between the fractional mean-field BSDEs and PDEs.

Generalized Mean-Field Fractional BSDEs With Non-Lipschitz Coefficients

In this paper we consider one dimensional generalized mean-field backward stochastic differential equations (BSDEs) driven by fractional Brownian motion, i.e., the generators of our mean-field FBSDEs depend not only on the solution but also on the law of the solution. We first give a totally new comparison theorem for such type of BSDEs under Lipschitz condition. Furthermore, we study the existence of the solution of such mean-field FBSDEs when the coefficients are only continuous and with a linear growth.

Mean-field backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients

In this paper, we investigate mean-field backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients. The existence and uniqueness theorem are obtained via a monotone convergence argument and a linearization method. Moreover, we establish the corresponding comparison theorem.

Controllability Gramian and Kalman rank condition for mean-field control systems

This paper is concerned with the exact controllability of linear mean-field stochastic systems with deterministic coefficients. With the help of the theory of mean-field backward stochastic differential equations (MF-BSDEs, for short) and some delicate analysis, we obtain a mean-field version of the Gramian matrix criterion for the general time-variant case, and a mean-field version of the Kalman rank condition for the special time-invariant case.

Maximum principle for infinite horizon optimal control of mean-field backward stochastic systems with delay and noisy memory

In this paper, we consider a problem of optimal control of an infinite horizon mean-field backward stochastic differential equation with delay and noisy memory under partial information. We derive necessary and sufficient maximum principles using Malliavin calculus technique for such a system. A class of mean-field time-advanced stochastic differential equations is introduced as the adjoint process which involves partial derivatives of the Hamiltonian functions and their Malliavin derivatives. To illustrate our theoretical results, we give an example for a linear-quadratic mean-field backward delay stochastic system with noisy memory on infinite horizon to obtain the optimal control. Also, we apply our results to pension fund problems with delay and noisy memory which are arising from the financial market.

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.

Lp solution of general mean-field BSDEs with continuous coefficients

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution (Y, Z) but also on the law PY of Y. The first part of the paper is devoted to the existence and uniqueness of solutions in Lp, 1 < p ≤ 2, where the monotonicity conditions are satisfied. Next, we show that if the generator f is uniformly continuous in (μ, y, z), uniformly with respect to (t, ω), and if the terminal value ξ belongs to Lp(Ω, F, P) with 1 < p ≤ 2, the mean-field BSDE has a unique Lp solution.

Anticipated mean-field backward stochastic differential equations with jumps∗

In this paper, we prove an existence and uniqueness theorem and a comparison theorem for a class of anticipated mean-field backward stochastic differential equations with jumps.

Generalized mean-field backward stochastic differential equations and related partial differential equations

In this paper, we derive the existence and uniqueness of the solution for a new class of mean-field backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. In addition, we study the generalized mean-field backward stochastic differential equations in a Markovian framework, that is, it is associated with a reflected McKean-Vlasov stochastic differential equation. This allows us to give a probabilistic formula for viscosity solution of a nonlocal partial differential equation with a nonlinear Neumann boundary condition.

A kind of non-zero sum mixed differential game of backward stochastic differential equation

This paper is concerned with a non-zero sum mixed differential game problem described by a backward stochastic differential equation. Here the term “mixed” means that this game problem contains a deterministic control v_{1} of Player 1 and a random control process v_{2} of Player 2. By virtue of the classical variational method, a necessary condition and an Arrow’s sufficient condition for the mixed stochastic differential game problem are presented. A linear–quadratic mixed differential game problem is discussed, and the corresponding Nash equilibrium point is explicitly expressed by the solution of mean-field forward–backward stochastic differential equation. The most distinguishing feature, compared with the existing literature, is that the optimal state process of the linear–quadratic game satisfies a linear mean-field backward stochastic differential equation. Finally, a home mortgage and wealth management problem is given to illustrate our theoretical results.

A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

In this paper we prove a maximum principle of optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps in the case where the diffusion coefficients depend on control, the control set does not need to be convex, the coefficients of jump terms are independent of control as well as the coefficients of mean-field backward stochastic differential equations depend on the joint law of (X(t), Y (t)). Since the coefficients depend on measure, higher mean-field terms could be involved. In order to analyse them, two new adjoint equations are brought in and several new generic estimates of their solutions are investigated. Utilizing these subtle estimates, the second-order expansion of the cost functional, which is the key point to analyse the necessary condition, is obtained, and where after the stochastic maximum principle. An illustrative application to mean-field game is considered.

An asymmetric information non-zero sum differential game of mean-field backward stochastic differential equation with applications

This paper is concerned with a kind of non-zero sum differential game driven by mean-field backward stochastic differential equation (MF-BSDE) with asymmetric information, whose novel feature is that both the state equation and the cost functional are of mean-field type. A necessary condition and a sufficient condition for Nash equilibrium point of the above problem are established. As applications, a mean-field linear-quadratic (MF-LQ) problem and a financial problem are studied.

Mean‐field backward stochastic differential equation with non‐Lipschitz coefficient

AbstractThis paper establishes a new existence and uniqueness result of a solution for one dimensional mean‐field backward stochastic differential equation (MFBSDE), where its coefficient is weaker than the classical Lipschitz case. An example is given to illustrate its applicability. This new solution will provide a key tool for studying mean‐field control problems.

Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application

This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. Then the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high-dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.

Singular Control Optimal Stopping of Memory Mean-Field Processes

The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory; (ii) reflected advanced mean-field backward stochastic differential equations; and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: (1) We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations; (2) we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information; and (3) we deduce a relation between the optimal singular control of an MMSDE and the optimal stopping of such processes.