Special Session 29: Mean field stochastic control problems and related topics

Optimal control problems with generalized mean-field dynamics and viscosity solution to Master Bellman equation

Rainer Buckdahn Universite de Bretagne Occidentale France Co-Author(s):

Abstract: In this talk we study an optimal control problem of generalized mean-field dynamics with open loop controls, where the coefficients depend not only on the state processes and controls, but also on the joint law of them. The value function $V$ defined in a conventional way, but it does not satisfy the Dynamic Programming Principle (DPP for short). For this reason we introduce subtly a novel value function $\vartheta$, which is closely related to the original value function $V$, such that, a description of $\vartheta$, as a solution of a partial differential equation (PDE), also characterizes $V$. We establish the DPP for $\vartheta$. By using an intrinsic notion of viscosity solutions, we show that the value function $\vartheta$ is a viscosity solution to a Master Bellman equation on a subset of Wasserstein space of probability measures. The uniqueness of viscosity solution is proved for coefficients which depend on the time and the joint law of the control process and the controlled process. The talk is based on joint work with Juan Li (SDU, China), Zhanxin Li (SDU, China).

Stochastic PDEs driven by $G-$Brownian motion and the associated Backward Doubly Stochastic Differential Equations

Laurent Denis Le Mans University France Co-Author(s):

Abstract: We study the well-posedness of quasilinear stochastic partial differential equations driven by $G-$Brownian motion (GSPDEs for short) and the associated backward doubly stochastic differential equations (BDSDEs for short). We first prove the existence and uniqueness of weak solution to GSPDEs by analytical approach, and then solve the corresponding BDSDEs. Finally, we establish the relation between GSPDEs and BDSDEs.

Doubly Reflected Backward SDEs Driven by G-Brownian Motions and Fully Nonlinear PDEs with Double Obstacles

Hanwu Li Shandong University Peoples Rep of China Co-Author(s):    Ning Ning

Abstract: In this talk, we introduce a new method to study the doubly reflected backward stochastic differential equation driven by G-Brownian motion (G-BSDE). Our approach involves approximating the solution through a family of penalized reflected G-BSDEs with a lower obstacle that are monotone decreasing. By employing this approach, we establish the well-posedness of the solution of the doubly reflected G-BSDE with the weakest known conditions, and uncover its relationship with the fully nonlinear partial differential equation with double obstacles for the first time.

Mean field stochastic control problems under sublinear expectation

Juan Li Shandong University Peoples Rep of China Co-Author(s):

Abstract: In this talk we study Pontryagin`s stochastic maximum principle for a mean-field optimal control problem under Peng`s $G$-expectation. The dynamics of the controlled state process is given by a stochastic differential equation driven by a $G$-Brownian motion, whose coefficients depend not only on the control, the controlled state process but also on its law under the $G$-expectation. Also the associated cost functional is of mean-field type. Under the assumption of a convex control state space we study the stochastic maximum principle, which gives a necessary optimality condition for control processes. Under additional convexity assumptions on the Hamiltonian it is shown that this necessary condition is also a sufficient one. The main difficulty which we have to overcome in our work consists in the differentiation of the $G$-expectation of parameterized random variables. Based on a joint work with Rainer Buckdahn (UBO, France), Bowen He (SDU, China).

Mean Field Games of Major-Minor Agents with Recursive Functionals

Wenqiang Li Shandong University Peoples Rep of China Co-Author(s):

Abstract: This paper studies a general class of mean field games involving a major agent and a large number of minor agents, whose payoff functionals are \emph{recursive} and represented in terms of the solution of backward stochastic differential equations, referred to as recursive major-minor (RMM) problems. Our RMM modeling encompasses weak couplings of empirical averages into the recursive functionals and dynamics of both major and minor agents and incorporates general non-additive functionals. The auxiliary limiting game of RMM is constructed via a novel mixed triple-agent leader-follower-Nash games. The associated consistency system is derived and related asymptotic major-minor equilibrium is constructed. In addition, linear-quadratic settings of RMM problems are studied to illustrate our results.

Fractional BSPDEs with Applications to Optimal Control of Partially Observed Systems with Jumps

Yunzhang Li Fudan University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider the Cauchy problem for backward stochastic partial differential equations (BSPDEs) involving fractional Laplacian operator. By employing the martingale representation theorem and the fractional heat kernel, we construct an explicit form of the solution, thereby demonstrating the existence and uniqueness of a strong solution in Holder space. Utilizing the freezing coefficients method as well as the continuation method, we establish Holder estimates for general BSPDEs with random coefficients dependent on space time variables. As an application, we use the fractional adjoint equation to study stochastic optimal control of the partially observed systems driven by Levy processes. This work is jointed with Yuyang Ye and Shanjian Tang.

On Some Generic Properties of Mean-Field Stochastic Differential Equations

Brahim Mezerdi King Fahd University of Petroleum and Minerals Saudi Arabia Co-Author(s):

Abstract: We investigate Mean-Field Stochastic Differential Equations (MFSDEs), where the coefficients depend on both the state and the marginal distribution of the solution. Under the assumption of global Lipschitz continuity of the coefficients in both arguments, the existence and uniqueness of a strong solution are well-established. This paper addresses the topological structure of the set of continuous coefficients that yield unique strong solutions and convergent successive approximations. We prove that this set is residual within the Baire space of uniformly continuous functions, implying its genericity in a topological sense. Moreover, we establish the generic property of convergence of Picard successive approximations and the Euler numerical scheme using similar techniques.

The smallest singular value of random matrices

Hanchao Wang Shandong University Peoples Rep of China Co-Author(s):

Abstract: There has been much research activity around extreme singular values of random matrices in probability, geometric functional analysis, mathematical physics, and other fields. In this talk, I will present our recent result on estimating the smallest singular value of a random subexponential matrix. This talk is based on recent joint works with Guozheng Dai and Zhonggen Su.

Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations with a Type of Random Coefficients

Qingmeng Wei Northeast Normal Univeristy Peoples Rep of China Co-Author(s):    Hongwei Mei, Jiongmin Yong

Abstract: Motivated by linear-quadratic optimal control problems (LQ problems, for short) for mean-field stochastic differential equations (SDEs, for short) with the coefficients containing regime switching governed by a Markov chain, we consider an LQ problem for an SDE with the coefficients being adapted to a filtration independent of the Brownian motion driving the control system. Classical approach of completing the square is applied to the current problem and obvious shortcomings are indicated. Open-loop and closed-loop solvability are introduced and characterized.

Path-dependent controlled mean-field coupled forward-backward SDEs. The associated stochastic maximum principle

Chuanzhi Xing Shandong University Peoples Rep of China Co-Author(s):

Abstract: In the present paper we discuss a new type of mean-field coupled forward-backward stochastic differential equations (MFFBSDEs). The novelty consists in the fact that the coefficients of both the forward as well as the backward SDEs depend not only on the controlled solution processes $(X_t,Y_t,Z_t)$ at the current time $t$, but also on the law of the paths of $(X,Y,u)$ of the solution process and the process by which it is controlled. The existence for such a MFFBSDE which is fully coupled through the law of the paths of $(X,Y)$ in the coefficients of both the forward and the backward equations is proved under rather general assumptions. The main part of the work is devoted to the study of Pontryagin`s maximal principle for such a MFFBSDE. The dependence of the coefficients of the law of the paths of the solution processes and their control makes that a completely new and interesting criterion for the optimality of a stochastic control for the MFFBSDE is obtained. Furthermore, we show that this necessary optimality condition is, under the assumption of convexity of the Hamiltonian, also sufficient. The talk is based on joint work with Rainer Buckdahn (UBO, France), Juan Li (SDU, China), Junsong Li (SDU, China).

Exact Controllability for Linear Stochastic Game-Based Control Systems

Zhiyong Yu Shandong University Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with the exact controllability of a linear stochastic game-based control system (SGBCS) with time-variant random coefficients. A Gram-type criterion is obtained. At the same time, the equivalence between the exact controllability of this SGBCS and the exact observability of a dual system is established. Moreover, an admissible control that can steer the state from any initial vector to any terminal random variable is constructed in closed form.

Backward Stochastic Partial Differential Equations with Conormal Boundary Conditions

Jing Zhang Fudan University Peoples Rep of China Co-Author(s):

Abstract: We prove the existence and uniqueness of strong solution to backward stochastic partial differential equations (BSPDEs for short) with conormal boundary conditions in high dimensional case. We apply our results to the linear-quadratic optimal control problems for stochastic partial differential equations (SPDEs for short) and obtain a maximum principle of Pontryagin`s type. This is a joint work with Jinniao Qiu (University of Calgary, Canada) and Xue Yang (Tianjin University, China).

Some New Results on Entropy Regularized Backward Stochastic Control Systems

Qi Zhang Fudan University Peoples Rep of China Co-Author(s):    Ziyue Chen, Qi Zhang

Abstract: The entropy regularization is inspired by information entropy from machine learning and the ideas of exploration and exploitation in reinforcement learning, which appears in the control problem to design an approximating algorithm for the optimal control. I will introduce our new results on the optimal exploratory control for backward stochastic system, generated by the backward stochastic differential equation and with the entropy regularization in its cost functional. We give the theoretical depict of the optimal relaxed control so as to lay the foundation for the application of such a backward stochastic control system to mathematical finance and algorithm implementation.