Comparison theorems for multi-dimensional BSDEs with jumps and applications to constrained stochastic linear-quadratic control

In this paper, we, for the first time, establish two comparison theorems for multi-dimensional backward stochastic differential equations with jumps. Our approach is novel and completely different from the existing results for one-dimensional case. Using these and other delicate tools, we then construct solutions to coupled two-dimensional stochastic Riccati equation with jumps in both standard and singular cases. In the end, these results are applied to solve a cone-constrained stochastic linear-quadratic and a mean-variance portfolio selection problem with jumps. Different from no jump problems, the optimal (relative) state processes may change their signs, which is of course due to the presence of jumps

Modeling and computation of an integral operator Riccati equation for an infinite-dimensional stochastic differential equation governing streamflow discharge

We propose a linear-quadratic (LQ) control problem of streamflow discharge by optimizing an infinite-dimensional jump-driven stochastic differential equation (SDE). Our SDE is a superposition of Ornstein-Uhlenbeck processes (supOU process), generating a sub-exponential autocorrelation function observed in actual data. The integral operator Riccati equation is heuristically derived to determine the optimal control of the infinite-dimensional system. In addition, its finite-dimensional version is derived with a discretized distribution of the reversion speed and computed by a finite difference scheme. The optimality of the Riccati equation is analyzed by a verification argument. The supOU process is parameterized based on the actual data of a perennial river. The convergence of the numerical scheme is analyzed through computational experiments. Finally, we demonstrate the application of the proposed model to realistic problems along with the Kolmogorov backward equation for the performance evaluation of controls

Optimal and Robust Control for a Class of Nonlinear Stochastic Systems

This thesis focuses on theoretical research of optimal and robust control theory for a class of nonlinear stochastic systems. The nonlinearities that appear in the diffusion terms are of a square-root type. Under such systems the following problems are investigated: optimal stochastic control in both finite and infinite horizon; robust stabilization and robust H∞ control; H₂/H∞ control in both finite and infinite horizon; and risk-sensitive control. The importance of this work is that explicit optimal linear controls are obtained, which is a very rare case in the nonlinear system. This is regarded as an advantage because with explicit solutions, our work becomes easier to be applied into the real problems. Apart from the mathematical results obtained, we have also introduced some applications to finance

Advertising for a new product introduction: a stochastic approach

Molte politiche di marketing possono essere correttamente spiegate e analizzate solo grazie ad un approccio stocastico al problema. In questo lavoro la pianificazione di una campagna pubbliciaria, che precede l'introduzione di un prodotto nel mercato, e' stata studiata utilizzando il controllo stocastico e sfruttando alcuni risultati recenti di controllo stocastico lineare quadratico

A stochastic linear-quadratic optimal control problem with jumps in an infinite horizon

In this paper, a stochastic linear-quadratic (LQ, for short) optimal control problem with jumps in an infinite horizon is studied, where the state system is a controlled linear stochastic differential equation containing affine term driven by a one-dimensional Brownian motion and a Poisson stochastic martingale measure, and the cost functional with respect to the state process and control process is quadratic and contains cross terms. Firstly, in order to ensure the well-posedness of our stochastic optimal control of infinite horizon with jumps, the $L^2$-stabilizability of our control system with jump is introduced. Secondly, it is proved that the $L^2$-stabilizability of our control system with jump is equivalent to the non-emptiness of the admissible control set for all initial state and is also equivalent to the existence of a positive solution to some integral algebraic Riccati equation (ARE, for short). Thirdly, the equivalence of the open-loop and closed-loop solvability of our infinite horizon optimal control problem with jumps is systematically studied. The corresponding equivalence is established by the existence of a $stabilizing\ solution$ of the associated generalized algebraic Riccati equation, which is different from the finite horizon case. Moreover, any open-loop optimal control for the initial state $x$ admiting a closed-loop representation is obatined

On the Value of Linear Quadratic Zero-sum Difference Games with Multiplicative Randomness: Existence and Achievability

We consider a wireless networked control system (WNCS) with multiple controllers and multiple attackers. The dynamic interaction between the controllers and the attackers is modeled as a linear quadratic (LQ) zero-sum difference game with multiplicative randomness induced by the multiple-input and multiple-output (MIMO) wireless fading channels of the controllers and attackers. We focus on analyzing the existence and achievability of the value of the zero-sum game. We first establish a general sufficient and necessary condition for the existence of the game value. This condition relies on the solvability of a modified game algebraic Riccati equation (MGARE) under an implicit concavity constraint, which is generally difficult to verify due to the intermittent controllability or almost sure uncontrollability caused by the multiplicative randomness. We then introduce a new positive semidefinite (PSD) kernel decomposition method induced by multiplicative randomness, through which we obtain a closed-form tight verifiable sufficient condition. Under the existence condition, we finally construct a saddle-point policy that is able to achieve the game value in a certain class of admissible policies. We demonstrate that the proposed saddle-point policy is backward compatible to the existing strictly feedback stabilizing saddle-point policy.Comment: 32 pages, 3 figure

Infinite Horizon Mean-Field Linear Quadratic Optimal Control Problems with Jumps and the related Hamiltonian Systems

In this work, we focus on an infinite horizon mean-field linear-quadratic stochastic control problem with jumps. Firstly, the infinite horizon linear mean-field stochastic differential equations and backward stochastic differential equations with jumps are studied to support the research of the control problem. The global integrability properties of their solution processes are studied by introducing a kind of so-called dissipation conditions suitable for the systems involving the mean-field terms and jumps. For the control problem, we conclude a sufficient and necessary condition of open-loop optimal control by the variational approach. Besides, a kind of infinite horizon fully coupled linear mean-field forward-backward stochastic differential equations with jumps is studied by using the method of continuation. Such a research makes the characterization of the open-loop optimal controls more straightforward and complete.Comment: 27page

Optimal Control of a Large Space Telescope Using an Annular Momentum Control Device

Application of a new development in the field of momentum storage devices, the Annular Momentum Control Device (AMCD), to the twin problems of large angle maneuvers and fine pointing control is considered. The basic concept of the AMCD consists of a spinning rim, with no central hub area, suspended by a minimum of three magnetic bearings, and driven by a noncontacting electromagnetic spin motor. The dissertation considers in detail the design of an optimal controller to achieve both large angle maneuvers and the fine pointing control of a Large Telescope (LST) with a single configuration, consisting of a single AMCD mounted in a single gimbal