/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components $Y$ and $Z$. We establish various sufficient conditions for the well-posedness of an ODE that dominates the characteristic BSDE, which leads to the existence of the desired regular decoupling random field, whence the solvability of the original FBSDE. A synthetic analysis of the solvability is given, as a "User's Guide," for a large class of FBSDEs that are not covered by the existing methods. Some of them have important implications in applications.

On well-posedness of forward-backward SDEs-A unified approach

A general maximum principle for optimal control of forward–backward stochastic systems

We identify a class of feedforward nonlinear systems that are linearizable by a coordinate change. Then we develop explicit expressions for the Lyapunov-based integrator forwarding recursive procedure of Sepulchre, Jankovic, and Kokotovic, which has its roots in a coordinate transformation proposed by Mazenc and Praly. The explicit expressions that we develop allow us to also find closed-form control laws for several classes of systems that are not feedback linearizable, including some that are in the feedforward form and others that are in what we refer to as the “block-feedforward” form. Performance advantages of Lyapunov-based forwarding controllers over nested saturation controllers have been well illustrated in the literature on examples. The analytical expressions for the Lyapunov functions and the control laws allow us to give quantitative performance bounds.

/pdf/feedback-linearizability-and-explicit-integrator-forwarding-40sf28qmcn.pdf

Feedback Linearizability and Explicit Integrator Forwarding Controllers for Classes of Feedforward Systems - eScholarship

In this paper, finite-time stability and stabilization problems for a class of linear stochastic systems are studied. First, a new concept of finite-time stochastic stability is defined for linear stochastic systems. Then, based on matrix inequalities, some sufficient conditions under which the stochastic systems are finite-time stochastically stable are given. Subsequently, the finite-time stochastic stabilization is studied and some sufficient conditions for the existence of a state feedback controller and a dynamic output feedback controller are presented by using a matrix inequality approach. An algorithm is given for solving the matrix inequalities arising from finite-time stochastic stability (stabilization). Finally, two examples are employed to illustrate the results.

Finite-Time Stability and Stabilization of Linear Itô Stochastic Systems with State and Control-Dependent Noise

This technical note is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by backward stochastic differential equations (BSDEs). This kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities. We establish a necessary condition and a sufficient condition in the form of maximum principle for open-loop equilibrium point of the foregoing games respectively. To explain the theoretical results, we use them to study a financial problem.

A Pontryagin's Maximum Principle for Non-Zero Sum Differential Games of BSDEs with Applications

The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls

A partial information non-zero sum differential game of backward stochastic differential equations with applications

In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraint for the cost functional described by the solution of a reflected backward stochastic differential equation. We give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equation.

Dynamic Programming Principle for One Kind of Stochastic Recursive Optimal Control Problem and Hamilton-Jacobi-Bellman Equation

A time-inconsistent stochastic optimal control problem with a recursive cost functional is studied. Equilibrium strategy is introduced, which is time-consistent and locally approximately optimal. By means of multiperson hierarchical differential games associated with partitions of the time interval, a family of approximate equilibrium strategy is constructed, and by sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB) equation is derived through which the equilibrium value function can be identified and the equilibrium strategy can be obtained. Moreover, a well-posedness result of the equilibrium HJB equation is established under certain conditions, and a verification theorem is proved. Finally, an illustrative example is presented, and some comparisons of different definitions of equilibrium strategy are put in order.

Zhiyong Yu

Papers

A Pontryagin's Maximum Principle for Non-Zero Sum Differential Games of BSDEs with Applications

The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls

A partial information non-zero sum differential game of backward stochastic differential equations with applications

Dynamic Programming Principle for One Kind of Stochastic Recursive Optimal Control Problem and Hamilton-Jacobi-Bellman Equation

Time-Inconsistent Recursive Stochastic Optimal Control Problems