Mean-Field, Infinite Horizon, Optimal Control of Nonlinear Stochastic Delay System Governed by Teugels Martingales Associated with Lévy Processes

This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.

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

This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels mar-tingales associated with Lévy processes.In either case,we obtain the optimality system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differen-tial equations.Moreover,the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.

A Mean-Field Optimal Control for Fully Coupled Forward-Backward Stochastic Control Systems with Lévy Processes

This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.

Generalized Backward Doubly Stochastic Differential Equations Driven by Lévy Processes with Continuous Coefficients

A new class of generalized backward doubly stochastic differential equations （GBDSDEs in short） driven by Teugels martingales associated with Levy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions.

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a family of pairwise strongly orthonormal martingales associated with L6vy processes （see e.g., Nualart and Schoutens＇ paper in 2000）. We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria （or backward linear-quadratic problem, or BLQ problem for short） is discussed and characterized by a stochastic Hamilton system.

Necessary and sufficient conditions for optimal control of stochastic systems associated with Lvy processes

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables, As an application, the linear quadratic stochastic optimal control problem is studied.

Reflected and Doubly Reflected BSDEs for L'evy Processes:Solutions and Comparison

In this paper we study reflected and doubly reflected backward stochastic differential equations （BSDEs, for short） driven by Teugels martingales associated with L^vy process satisfying some moment condi- tions and by an independent Brownian motion. For BSDEs with one reflecting barrier, we obtain a comparison theorem using the Tanaka-Meyer formula. For BSDEs with two reflecting barriers, we first prove the existence and uniqueness of the solutions under the Mokobodski＇s condition by using the Snell envelope theory and then we obtain a comparison result.

Forward–Backward SDEs Driven by Levy Process in Stopping Time Duration

As the first part in the present paper,we study a class of backward stochastic differential equation(BSDE,for short)driven by Teugels martingales associated with some Levy processes having moment of all orders and an independent Brownian motion.We obtain an existence and uniqueness result for this type of BSDEs when the final time is allowed to be random.As the second part,we prove,under a monotonicity condition,an existence and uniqueness result for fully coupled forward-backward stochastic differential equation(FBSDE,for short)driven by Teugels martingales in stopping time duration.As an illustration of our theoretical results,we deal with a portfolio selection in Levy-type market.