Journal ArticleDOI
Stochastic Control for Linear Systems Driven by Fractional Noises
Yaozhong Hu,Xun Yu Zhou +1 more
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TLDR
Optimal control and optimal value of the model are explicitly obtained based on the solution to a new Riccati-type equation involving both FBM and normal Brownian motion.Abstract:
This paper is concerned with optimal control of stochastic linear systems involving fractional Brownian motion (FBM). First, as a prerequisite for studying the underlying control problems, some new results on stochastic integrals and stochastic differential equations associated with FBM are established. Then, three control models are formulated and studied. In the first two models, the state is scalar-valued and the control is taken as Markovian. Either the problems are completely solved based on a Riccati equation (for model 1, where the cost is a quadratic functional on state and control variables) or optimality is characterized (for model 2, where the cost is a power functional). The last control model under investigation is a general one, where the system involves the Stratonovich integral with respect to FBM, the state is multidimensional, and the admissible controls are not limited to being Markovian. A new Riccati-type equation, which is a backward stochastic differential equation involving both FBM and normal Brownian motion, is introduced. Optimal control and optimal value of the model are explicitly obtained based on the solution to this Riccati-type equation.read more
Citations
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Journal ArticleDOI
Backward Stochastic Differential Equation Driven by Fractional Brownian Motion
Yaozhong Hu,Shige Peng +1 more
TL;DR: In the nonlinear case, the existence and uniqueness of the solutions are obtained under some mild assumptions and an inequality of the type similar to in the classical backward stochastic differential equations leads to a fixed point principle.
Journal ArticleDOI
Linear-Quadratic Fractional Gaussian Control
TL;DR: In this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean, square integrable Stochastic process with continuous sample paths and a cost functional that is quadratic in the system state is solved.
Journal ArticleDOI
Robust Stabilization of Uncertain Time-Delay Systems With Fractional Stochastic Noise Using the Novel Fractional Stochastic Sliding Approach and Its Application to Stream Water Quality Regulation
TL;DR: Stochastic systems with fractional Gaussian noise (fGn) are stochastically stabilized using a new robust sliding mode control scheme and a fractional Ito process is proposed which is proven to be attainable almost surely in finite time.
Journal ArticleDOI
Integral sliding mode control for robust stabilisation of uncertain stochastic time-delay systems driven by fractional Brownian motion
TL;DR: By applying the proposed fractional infinitesimal operator, the sufficient robust stability conditions are derived in the form of linear matrix inequalities and the proposed method guarantees the reachability of the sliding surface in finite time, and the closed-loop system will be stable in probability for all Hurst indices of the fBm in the range of .
Journal ArticleDOI
Prediction for some processes related to a fractional Brownian motion
TL;DR: In this article, explicit expressions are given for conditional expectations for the prediction of some stochastic processes that are obtained from a fractional Brownian motion with the Hurst parameter in the interval (0, 1).
References
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Book
Stochastic differential equations and diffusion processes
TL;DR: In this article, Stochastic Differential Equations and Diffusion Processes are used to model the diffusion process in stochastic differential equations. But they do not consider the nonlinearity of diffusion processes.
Journal ArticleDOI
Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00
Book
Stochastic controls : Hamiltonian systems and HJB equations
Jiongmin Yong,Xun Yu Zhou +1 more
TL;DR: In this article, the authors consider the problem of deterministic control problems in the context of stochastic control systems and show that the optimal control problem can be formulated in a deterministic manner.
Journal ArticleDOI
Stochastic Analysis of the Fractional Brownian Motion
TL;DR: In this article, the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations was proved for the Ito formula.
Journal ArticleDOI
Fractional white noise calculus and applications to finance
TL;DR: In this paper, a fractional white noise calculus was developed for markets modeled by the Ito type of stochastic differential equations driven by fractional Brownian motion BH(t).