Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Risk-Sensitive Mean-Field Type Control under Partial Observation

We establish a stochastic maximum principle (SMP) for control problems of partially observed diffusions of mean-field type with risk-sensitive performance functionals.Comment: arXiv admin note: text overlap with arXiv:1404.144

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 $\epsilon$-Nash equilibrium properties are also presented.Comment: 19 page

A Risk-Sensitive Global Maximum Principle for Controlled Fully Coupled FBSDEs with Applications

This paper is concerned with a kind of risk-sensitive optimal control problem for fully coupled forward-backward stochastic systems. The control variable enters the diffusion term of the state equation and the control domain is not necessarily convex. A new global maximum principle is obtained without assuming that the value function is smooth. The maximum condition, the first- and second-order adjoint equations heavily depend on the risk-sensitive parameter. An optimal control problem with a fully coupled linear forward-backward stochastic system and an exponential-quadratic cost functional is discussed. The optimal feedback control and optimal cost are obtained by using Girsanov's theorem and completion-of-squares approach via risk-sensitive Riccati equations. A local solvability result of coupled risk-sensitive Riccati equations is given by Picard-Lindel$\ddot{o}$f's Theorem.Comment: 31 page