DISCRETE TIME STOCHASTIC EQUILIBRIUM WITH INFINITE HORIZON INCOMPLETE ASSET MARKETS

This paper examines the existence of general equilibrium in a discrete time economy with the infinite horizon incomplete markets.There is a single good at each node in the event tree.The existence of general equilibrium for the infinite horizon economy is proved by taking limit of equilibria in truncated economies in which trade stops at a sequence of dates.

Equilibria Immune to Deviations by Coalitions in Infinite Horizon Non-Cooperative Games

Infinite horizon discrete time non-cooperative games with observable actions of players and discounting of future single period payoffs are a suitable tool for analyzing emergence and sustainability of cooperation between all players because they do not contain the last period. A subgame perfect equilibrium is a standard solution concept for them. It requires only immunity to unilateral deviations in any subgame. It does not address immunity to deviations by coalitions. In particular, it does not rule out cooperation based on punishments of unilateral deviations that the grand coalition would like to forgive. We first briefly review concepts of renegotiation-proofness that rule out such forgiveness. Then we discuss the concept of strong perfect equilibrium that requires immunity to all deviations by all coalitions in all subgames. In games with only one level of players (e.g. members of the population engaged in the same type of competitive activity), it fails to exist when the Pareto efficient frontier of the set of single period payoff vectors has no sufficiently large flat portion. In such a case, it is not possible to punish unilateral deviations in a weakly Pareto efficient way. In games with two levels of players (e.g. members of two populations with symbiotic relationship, while activities within each population are competitive), it is possible to overcome this problem. The sum of benefits of all players during a punishment can be the same as when nobody is punished but its distribution between the two populations can be altered in favor of the punishers.

An Infinite Horizon Linear Quadratic Problem with Unbounded Controls in Hilbert Space

An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered.The state weight operator is allowed to be inddefinite while the control weight operator is coercive.Under the exponential stabilization condition,it is proved that any optimal control and its optimal trajectory are continuous.The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established.The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation.

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.

Infinite Horizon Backward Doubly Stochastic Differential Equations with Non-degenerate Terminal Functions and Their Stationary Property

In this paper we consider infinite horizon backward doubly stochastic differential equations （BDS- DEs for short） coupled with forward stochastic differential equations, whose terminal functions are non-degenerate. For such kind of BDSDEs, we study the existence and uniqueness of their solutions taking values in weighted Lp（dx）¤L2（dx）space （p _〉 2）, and obtain the stationary property for the solutions.

Infinite Horizon Forward-Backward Doubly Stochastic Differential Equations and Related SPDEs

A type of infinite horizon forward-backward doubly stochastic differential equations is studied.Under some monotonicity assumptions,the existence and uniqueness results for measurable solutions are established by means of homotopy method.A probabilistic interpretation for solutions to a class of stochastic partial differential equations combined with algebra equations is given.A significant feature of this result is that the forward component of the FBDSDEs is coupled with the backward variable.

Solving infinite horizon nonlinear optimal control problems using an extended modal series method

This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's maximum principle,is transformed into a sequence of linear time-invariant TPBVPs.Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series.Hence,to obtain the optimal solution,only the techniques for solving linear ordinary differential equations are employed.An efficient algorithm is also presented,which has low computational complexity and a fast convergence rate.Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP.The results not only demonstrate the efficiency,simplicity,and high accuracy of the suggested approach,but also indicate its effectiveness in practical use.

Stability of networked control systems with multi-step delay based on time-division algorithm

A new control mode is proposed for a networked control system whose network-induced delay is longer than a sampling period. A time-division algorithm is presented to implement the control and for the mathematical modeling of such networked control system. The infinite horizon controller is designed, which renders the networked control system mean square exponentially stable.Simulation results show the validity of the proposed theory.

Distributed H2/H∞ Filter Design for Discrete-Time Switched Systems

This paper addresses an infinite horizon distributed H2/H∞ filtering for discrete-time systems under conditions of bounded power and white stochastic signals. The filter algorithm is designed by computing a pair of gains namely the estimator and the coupling. Herein, we implement a filter to estimate unknown parameters such that the closed-loop multi-sensor accomplishes the desired performances of the proposed H2 and H∞ schemes over a finite horizon. A switched strategy is implemented to switch between the states once the operation conditions have changed due to disturbances. It is shown that the stability of the overall filtering-error system with H2/H∞ performance can be established if a piecewise-quadratic Lyapunov function is properly constructed. A simulation example is given to show the effectiveness of the proposed approach.

G-forward performance process and representation of homothetic case via ergodic quadratic G-BSDE

We introduce a new type of robust forward criterion under model uncertainty,called the G-forward performance process,which extends the classical notion of forward performance process to the G-expectation framework.We then derive the representations of homothetic G-forward performance processes in a single stochastic factor model with uncertainty,building on the well-posedness of ergodic and infinite horizon backward stochastic differential equations driven by G-Brownian motion(G-BSDEs)with quadratic generators.

Coordinating Pricing and Inventory Control in a Fluctuating Environment

This paper addresses the simultaneous determination of pricing and inventory replenishment strate- gies under a fluctuating environment. Specifically, we analyze the single item, periodic review model. The demand consists of two parts： the deterministic component, which is influenced by the price, and the stochastic component （perturbation）. The distribution of the stochastic component is determined by the current state of an exogenous Markov chain. The price that is charged in any given period can be specified dynamically. A replenishment order may be placed at the beginning of some or all of the periods, and stockouts are fully backlogged. Ordering costs that are lower semicontinuous, and inventory/backlog （or surplus） costs that are continuous with polynomial growth. Finite-horizon and infinite-horizon problems are addressed. Existence of optimal policies is established. Furthermore, optimality of （s,S,p）-type policies is proved when the ordering cost consists of fixed and proportional cost components and the surplus cost （these costs are all state-dependent） is convex.

Model-free method for LQ mean-field social control problems with one-dimensional state space

This paper presents a novel model-free method to solve linear quadratic(LQ)mean-field control problems with one-dimensional state space and multiplicative noise.The focus is on the infinite horizon LQ setting,where the conditions for solution either stabilization or optimization can be formulated as two algebraic Riccati equations(AREs).The proposed approach leverages the integral reinforcement learning technique to iteratively solve the drift-coefficient-dependent stochastic ARE(SARE)and other indefinite ARE,without requiring knowledge of the system dynamics.A numerical example is given to demonstrate the effectiveness of the proposed algorithm.