arXiv subjects

Jianhui Huang

Publications and source records attributed to Jianhui Huang.

27 records · Page 2Linked to original sources

arXiv2015

Necessary Condition for Near Optimal Control of Linear Forward-backward Stochastic Differential Equations

This paper investigates the near optimal control for a kind of linear stochastic control systems governed by the forward backward stochastic differential equations, where both the drift and diffusion terms are allowed to depend on controls and the control domain is not assumed to be convex. In the previous work (Theorem 3.1) of the second and third authors [\textit{% Automatica} \textbf{46} (2010) 397-404], some problem of near optimal control with the control dependent diffusion is addressed and our current paper can be viewed as some direct response to it. The necessary condition of the near-optimality is established within the framework of optimality variational principle developed by Yong [\textit{SIAM J. Control Optim.} \textbf{48} (2010) 4119--4156] and obtained by the convergence technique to treat the optimal control of FBSDEs in unbounded control domains by Wu [% \textit{Automatica} \textbf{49} (2013) 1473--1480]. Some new estimates are given here to handle the near optimality. In addition, an illustrating example is discussed as well.

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arXiv2014

A Class of Mean-field LQG Games with Partial Information

The large-population system consists of considerable small agents whose individual behavior and mass effect are interrelated via their state-average. The mean-field game provides an efficient way to get the decentralized strategies of large-population system when studying its dynamic optimizations. Unlike other large-population literature, this current paper possesses the following distinctive features. First, our setting includes the partial information structure of large-population system which is practical from real application standpoint. Specially, two cases of partial information structure are considered here: the partial filtration case (see Section 2, 3) where the available information to agents is the filtration generated by an observable component of underlying Brownian motion; the noisy observation case (Section 4) where the individual agent can access an additive white-noise observation on its own state. Also, it is new in filtering modeling that our sensor function may depend on the state-average. Second, in both cases, the limiting state-averages become random and the filtering equations to individual state should be formalized to get the decentralized strategies. Moreover, it is also new that the limit average of state filters should be analyzed here. This makes our analysis very different to the full information arguments of large-population system. Third, the consistency conditions are equivalent to the wellposedness of some Riccati equations, and do not involve the fixed-point analysis as in other mean-field games. The $ε$-Nash equilibrium properties are also presented.

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arXiv2014

Mean Field Linear-Quadratic-Gaussian (LQG) Games of Forward-Backward Stochastic Differential Equations

This paper studies a new class of dynamic optimization problems of large-population (LP) system which consists of a large number of negligible and coupled agents. The most significant feature in our setup is the dynamics of individual agents follow the forward-backward stochastic differential equations (FBSDEs) in which the forward and backward states are coupled at the terminal time. This current paper is hence different to most existing large-population literature where the individual states are typically modeled by the SDEs including the forward state only. The associated mean-field linear-quadratic-Gaussian (LQG) game, in its forward-backward sense, is also formulated to seek the decentralized strategies. Unlike the forward case, the consistency conditions of our forward-backward mean-field games involve six Riccati and force rate equations. Moreover, their initial and terminal conditions are mixed thus some special decoupling technique is applied here. We also verify the $ε$-Nash equilibrium property of the derived decentralized strategies. To this end, some estimates to backward stochastic system are employed. In addition, due to the adaptiveness requirement to forward-backward system, our arguments here are not parallel to those in its forward case.

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arXiv2014

Mean Field Linear-Quadratic-Gaussian (LQG) Games: Major and Minor Players

This paper is concerned with a backward-forward stochastic differential equation (BFSDE) system, in which a large number of negligible agents are coupled in their dynamics via state average. Here some BSDE is introduced as the dynamics of major player, while dynamics of minor players are described by SDEs. Some auxiliary mean-field SDEs (MFSDEs) and a $3\times2$ mixed forward-backward stochastic differential equation (FBSDE) system are considered and analyzed instead of involving the fixed-point analysis as in other mean-field games. We also derive the decentralized strategies which are shown to satisfy the $ε$-Nash equilibrium property.

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arXiv2013

Mean Field Linear-Quadratic-Gaussian (LQG) Games for Stochastic Integral Systems

In this paper we discuss a class of mean field linear-quadratic-Gaussian (LQG) games for large population system which has never been addressed by existing literature. The features of our works are sketched as follows. First of all, our state is modeled by stochastic Volterra-type equation which leads to some new study on stochastic "integral" system. This feature makes our setup significantly different from the previous mean field games where the states always follow some stochastic "differential" equations. Actually, our stochastic integral system is rather general and can be viewed as natural generalization of stochastic differential equations. In addition, it also includes some types of stochastic delayed systems as its special cases. Second, some new techniques are explored to tackle our mean-field LQG games due to the special structure of integral system. For example, unlike the Riccati equation in linear controlled differential system, some Fredholm-type equations are introduced to characterize the consistency condition of our integral system via the resolvent kernels. Third, based on the state aggregation technique, the Nash certainty equivalence (NCE) equation is derived and the set of associated decentralized controls are verified to satisfy the $ε$-Nash equilibrium property. To this end, some new estimates of stochastic Volterra equations are developed which also have their own interests.

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arXiv2012

A Mixed Linear Quadratic Optimal Control Problem with a Controlled Time Horizon

A mixed linear quadratic (MLQ, for short) optimal control problem is considered. The controlled stochastic system consists of two diffusion processes which are in different time horizons. There are two control actions: a standard control action $u(\cdot)$ enters the drift and diffusion coefficients of both state equations, and a stopping time $τ$, a possible later time after the first part of the state starts, at which the second part of the state is initialized with initial condition depending on the first state. A motivation of MLQ problem from a two-stage project management is presented. It turns out that solving MLQ problem is equivalent to sequentially solve a random-duration linear quadratic (RLQ, for short) problem and an optimal time (OT, for short) problem associated with Riccati equations. In particular, the optimal cost functional can be represented via two coupled stochastic Riccati equations. Some optimality conditions for MLQ problem is therefore obtained using the equivalence among MLQ, RLQ and OT problems.

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arXiv2012

A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations in Infinite Horizon

A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.

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arXiv2012

Regularity properties for general HJB equations. A BSDE method

In this work we investigate regularity properties of a large class of Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in form of a stochastic control system which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE (RBSDE). More precisely, we prove that, firstly, the unique viscosity solution $V(t,x)$ of such a HJB equation over the time interval $[0,T],$ with or without an obstacle, and with terminal condition at time $T$, is jointly Lipschitz in $(t,x)$, for $t$ running any compact subinterval of $[0,T)$. Secondly, for the case that $V$ solves a HJB equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that $V(t,x)$ is jointly semiconcave in $(t,x)$. These results extend earlier ones by Buckdahn, Cannarsa and Quincampoix [1]. Our approach embeds their idea of time change into a BSDE analysis. We also provide an elementary counter-example which shows that, in general, for the case that $V$ solves a HJB equation with a lower obstacle the semi-concavity doesn't hold true.

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arXiv2011

A stochastic maximum principle for backward delayed system via advanced stochastic differential equation (ASDE)

The main contributions of this paper are three fold. First, our primary concern is to investigate a class of stochastic recursive delayed control problems which arise naturally with sound backgrounds but have not been well-studied yet. For illustration purpose, some concrete examples are also provided here. We derive the stochastic maximum principle of sufficient condition to the optimal control in both cases with and without control delay. Second, it is interesting that a new class of time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. To our best knowledge, such equations have never been discussed in literature although they possess more academic values besides the control study here. Some existence and uniqueness result to ASDEs is presented. Third, to illustrate our theoretical results, some dynamic optimization problems are discussed based on our stochastic maximum principles. In particular, the optimal controls are derived explicitly by solving the associated time-advanced ordinary differential equation (AODE), the counterpart of the ASDE in its deterministic setup.

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