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This paper investigates a linear quadratic stochastic optimal control (LQSOC) problem with partial information. Firstly, by introducing two Riccati equations and a backward stochastic differential equation (BSDE), we solve this LQSOC problem under standard positive semidefinite assumptions. Secondly, by means of a perturbation approach, we study open-loop solvability of this problem when the weighting matrices in the cost functional are indefinite. Thirdly, we investigate weak closed-loop solvability of this problem and prove the equivalence between open-loop and weak closed-loop solvabilities. Finally, we give an example to illustrate the way for obtaining a weak closed-loop optimal strategy.

This paper considers the distributed online bandit optimization problem with nonconvex loss functions over a time-varying digraph. This problem can be viewed as a repeated game between a group of online players and an adversary. At each round, each player selects a decision from the constraint set, and then the adversary assigns an arbitrary, possibly nonconvex, loss function to this player. Only the loss value at the current round, rather than the entire loss function or any other information (e.g. gradient), is privately revealed to the player. Players aim to minimize a sequence of global loss functions, which are the sum of local losses. We observe that traditional multi-point bandit algorithms are unsuitable for online optimization, where the data for the loss function are not all a priori, while the one-point bandit algorithms suffer from poor regret guarantees. To address these issues, we propose a novel one-point residual feedback distributed online algorithm. This algorithm estimates the gradient using residuals from two points, effectively reducing the regret bound while maintaining $\mathcal{O}(1)$ sampling complexity per iteration. We employ a rigorous metric, dynamic regret, to evaluate the algorithm's performance. By appropriately selecting the step size and smoothing parameters, we demonstrate that the expected dynamic regret of our algorithm is comparable to existing algorithms that use two-point feedback, provided the deviation in the objective function sequence and the path length of the minimization grows sublinearly. Finally, we validate the effectiveness of the proposed algorithm through numerical simulations.

This paper studies a competitive optimal portfolio selection problem in a model where the interest rate, the appreciation rate and volatility rate of the risky asset are all stochastic processes, thus forming a non-Markovian financial market. In our model, all investors (or agents) aim to obtain an above-average wealth at the end of the common investment horizon. This competitive optimal portfolio problem is indeed a non-zero stochastic differential game problem. The quadratic BSDE theory is applied to tackle the problem and Nash equilibria in suitable spaces are found. We discuss both the CARA and CRRA utility cases. For the CARA utility case, there are three possible scenarios depending on market and competition parameters: a unique Nash equilibrium, no Nash equilibrium, and infinite Nash equilibria. The Nash equilibrium is given by the solutions of a quadratic BSDE and a linear BSDE with unbounded coefficient when it is unique. Different from the wealth-independent Nash equilibria in the existing literature, the equilibrium in our paper is of feedback form of wealth. For the CRRA utility case, the issue is a bit more complicated than the CARA utility case. We prove the solvability of a new kind of quadratic BSDEs with unbounded coefficients. A decoupling technology is used to relate the Nash equilibrium to a series of 1-dimensional quadratic BSDEs. With the help of this decoupling technology, we can even give the limiting strategies for both cases when the number of agent tends to be infinite., Comment: 23 pages

A linear quadratic (LQ) stochastic optimization system involving large population, which is driven by forward-backward stochastic differential equation (FBSDE), is investigated in this paper. Agents cooperate with each other to minimize the so-called social objective, which is rather different from mean field (MF) game. Employing forward-backward person-by-person optimality principle, we derive an auxiliary LQ control problem by decentralized information. A decentralized strategy is obtained by virtue of an MF-type forward-backward stochastic differential equation consistency condition. Applying Riccati equation decoupling method, we solve the consistency condition system. We also verify the asymptotic social optimality in this framework., Comment: 30 pages

This investigation is dedicated to a two-player zero-sum stochastic differential game (SDG), where a cost function is characterized by a backward stochastic differential equation (BSDE) with a continuous and monotonic generator regarding the first unknown variable, which possesses immense applicability in financial engineering. A verification theorem by virtue of classical solution of derived Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation is given. The dynamic programming principle (DPP) and unique weak (viscosity) solvability of HJBI equation are formulated through comparison theorem for BSDEs with monotonic generators and stability of viscosity solution. Some new regularity properties of value function are presented. Finally, we propose three concrete examples, which are concerned with resp., classical, and viscosity solution of HJBI equation, as well as a financial application where an investor with a non-Lipschitzian Epstein-Zin utility deals with market friction to maximize her utility preference.

This paper studies a class of continuous-time linear quadratic (LQ) mean-field game problems. We develop two system transformation data-driven algorithms to approximate the decentralized strategies of the LQ mean-field games. The main feature of the obtained data-driven algorithms is that they eliminate the requirement on all system matrices. First, we transform the original stochastic system into an ordinary differential equation (ODE). Subsequently, we construct some Kronecker product-based matrices by the input/state data of the ODE. By virtue of these matrices, we implement a model-based policy iteration (PI) algorithm and a model-based value iteration (VI) algorithm in a data-driven fashion. In addition, we also demonstrate the convergence of these two data-driven algorithms under some mild conditions. Finally, we illustrate the practicality of our algorithms via two numerical examples.

This paper presents a novel value iteration (VI) algorithm for finding the optimal control for a kind of infinite-horizon stochastic linear quadratic (SLQ) problem with unknown systems. First, an off-line algorithm is estabilished to obtain the optimal feedback control of our problem. Then, based on the off-line algorithm, the VI-based model-free algorithm and its convergence proof is provided. The main feature of the model-free algorithm is that a stabilizing control is not needed to initiate the algorithm. Finally, we validate our results with a simulation example.

In this paper, we consider a linear-quadratic optimal control problem of mean-field stochastic differential equation with jump diffusion, which is also called as an MF-LQJ problem. Here, cost functional is allowed to be indefinite. We use an equivalent cost functional method to deal with the MF-LQJ problem with indefinite weighting matrices. Some equivalent cost functionals enable us to establish a bridge between indefinite and positive-definite MF-LQJ problems. With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati equations are further characterized. Optimal control of the indefinite MF-LQJ problem is represented as a state feedback via solutions of Riccati equations. As a by-product, the method provides a new way to prove the existence and uniqueness of solution to mean field forward-backward stochastic differential equation with jump diffusion (MF-FBSDEJ, for short), where existing methods in literature do not work. Some examples are provided to illustrate our results.

In this paper, we study an optimal control problem of linear backward stochastic differential equation (BSDE) with quadratic cost functional under partial information. This problem is solved completely and explicitly by using a stochastic maximum principle and a decoupling technique. By using the maximum principle, a stochastic Hamiltonian system, which is a forward-backward stochastic differential equation (FBSDE) with filtering, is obtained. By decoupling the stochastic Hamiltonian system, three Riccati equations, a BSDE with filtering, and a stochastic differential equation (SDE) with filtering are derived. We then get an optimal control with a feedback representation. An explicit formula for the corresponding optimal cost is also established. As illustrative examples, we consider two special scalar-valued control problems and give some numerical simulations.

This paper is concerned with a linear-quadratic (LQ) Stackelberg differential game with mixed deterministic and stochastic controls. Here in the game, the follower is a random controller which means that the follower can choose adapted random processes, while the leader is a deterministic controller which means that the leader can choose only deterministic time functions. An open-loop Stackelberg equilibrium solution is considered. First, an optimal control process of the follower is obtained by maximum principle of controlled stochastic differential equation (SDE), which is a linear functional of optimal state variable and control variable of the leader, via a classical Riccati equation. Then an optimal control function of the leader is got via a direct calculation of derivative of cost functional, by the solution to a system of mean-field forward-backward stochastic differential equations (MF-FBSDEs). And it is represented as a functional of expectation of optimal state variable, together with solutions to a two-point boundary value problem of ordinary differential equation (ODE), by a system consisting of two coupled Riccati equations. The solvability of this new system of Riccati equation is discussed., Comment: 14 pages

This paper is concerned with the stochastic linear quadratic Stackelberg differential game with overlapping information, where the diffusion terms contain the control and state variables. Here the term "overlapping" means that there are common part between the follower's and the leader's information, while they have no inclusion relation. Optimal controls of the follower and the leader are obtained by the stochastic maximum principle, the direct calculation of the derivative of the cost functional and stochastic filtering. A new system of Riccati equations is introduced to represent the state estimate feedback of the Stackelberg equilibrium strategy. A special solvable case is then studied and is applied to the continuous-time principal-agent problem., Comment: 43 pages

This paper is concerned with a leader-follower stochastic differential game with asymmetric information, where the information available to the follower is based on some sub-$\sigma$-algebra of that available to the leader. Such kind of game problem has wide applications in finance, economics and management engineering such as newsvendor problems, cooperative advertising and pricing problems. Stochastic maximum principles and verification theorems with partial information are obtained, to represent the Stackelberg equilibrium. As applications, a linear-quadratic leader-follower stochastic differential game with asymmetric information is studied. It is shown that the open-loop Stackelberg equilibrium admits a state feedback representation if some system of Riccati equations is solvable., Comment: 20 pages. The main content of this paper is presented by the first author in IMS-China International Conference on Statistics and Probability, Kunming, P.R. China, June 2015

This article is concerned with an optimal control problem derived by mean-field forward-backward stochastic differential equation with noisy observation, where the drift coefficients of the state equation and the observation equation are linear with respect to the state and its expectation. The control problem is different from the existing literature on optimal control for mean-field stochastic systems, and has more applications in mathematical finance, e.g., asset-liability management problem with recursive utility, systematic risk model. Using a backward separation method with a decomposition technique, two optimality conditions along with two coupled forward-backward optimal filters are derived. Several linear-quadratic optimal control problems for mean-field forward-backward stochastic differential equations are studied. Closed-form optimal solutions are explicitly obtained in detailed situations., Comment: 16 pages; This is a correction for the third version

This paper focuses on a kind of linear quadratic non-zero sum differential game driven by backward stochastic differential equation with asymmetric information, which is a natural continuation of Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352]. Different from Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352], novel motivations for studying this kind of game are provided. Some feedback Nash equilibrium points are uniquely obtained by forward-backward stochastic differential equations, their filters and the corresponding Riccati equations with Markovian setting., Comment: 19 Pages