Topological dual of $B_∞(I, ₁(X,Y))$ with application to stochastic systems on Hilbert space

N.U. Ahmed

Displaying similar documents to “Topological dual of $B_{\infty} (I, ₁ (X, Y))$ with application to stochastic systems on Hilbert space”

Viability theorems for stochastic inclusions

Michał Kisielewicz (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Sufficient conditions for the existence of solutions to stochastic inclusions $x_{t} - x_{s} \in \int_{s}^{t} F_{τ} (x_{τ}) d τ + \int_{s}^{t} G_{τ} (x_{τ}) d w_{τ} + \int_{s}^{t} \int_{I R ⁿ} H_{τ, z} (x_{τ}) ν ̃ (d τ, d z)$ beloning to a given set K of n-dimensional cádlág processes are given.

Stochastic integration of functions with values in a Banach space

J. M. A. M. van Neerven, L. Weis (2005)

Studia Mathematica

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Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process ${W_{H} (t)}_{t \in [0, T]}$ . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator...

Set-valued stochastic integrals and stochastic inclusions in a plane

Władysław Sosulski (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form $z_{s, t} \in φ_{s, t} + \int_{0}^{s} \int_{0}^{t} F_{u, v} (z_{u, v}) d u d v + \int_{0}^{s} \int_{0}^{t} G_{u, v} (z_{u, v}) d w_{u, v}$

Measure valued solutions for stochastic evolution equations on Hilbert space and their feedback control

N.U. Ahmed (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we consider a class of semilinear stochastic evolution equations on Hilbert space driven by a stochastic vector measure. The nonlinear terms are assumed to be merely continuous and bounded on bounded sets. We prove the existence of measure valued solutions generalizing some earlier results of the author. As a corollary, an existence result of a measure solution for a forward Kolmogorov equation with unbounded operator valued coefficients is obtained. The main result is...

Limiting average cost control problems in a class of discrete-time stochastic systems

Nadine Hilgert, Onesimo Hernández-Lerma (2001)

Applicationes Mathematicae

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We consider a class of $ℝ^{d}$ -valued stochastic control systems, with possibly unbounded costs. The systems evolve according to a discrete-time equation $x_{t + 1} = G ₙ (x_{t}, a_{t}) + ξ_{t}$ (t = 0,1,... ), for each fixed n = 0,1,..., where the $ξ_{t}$ are i.i.d. random vectors, and the Gₙ are given functions converging pointwise to some function $G_{\infty}$ as n → ∞. Under suitable hypotheses, our main results state the existence of stationary control policies that are expected average cost (EAC) optimal and sample path average cost (SPAC)...

Representation of operators on $L^{1}$ -spaces by nondirect product measures

Anzelm Iwanik (1982)

Colloquium Mathematicum

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Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality

N.U. Ahmed (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We consider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary...

A relaxation theorem for partially observed stochastic control on Hilbert space

N.U. Ahmed (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we present a result on relaxability of partially observed control problems for infinite dimensional stochastic systems in a Hilbert space. This is motivated by the fact that measure valued controls, also known as relaxed controls, are difficult to construct practically and so one must inquire if it is possible to approximate the solutions corresponding to measure valued controls by those corresponding to ordinary controls. Our main result is the relaxation theorem which...

Weak compactness in the space of operator valued measures $M_{b} a (Σ, (X, Y))$ and its applications

N.U. Ahmed (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{b a} (Σ, (X, Y))$ . This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{b a} (Σ, ₁ (X, Y))$ . This result has interesting applications in optimization and control theory as illustrated by several examples.

Some Results on Stochastic Porous Media Equations

Viorel Barbu, Giuseppe Da Prato, Michael Röckner (2008)

Bollettino dell'Unione Matematica Italiana

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Some recent results about nonnegative solutions of stochastic porous media equations in bounded open subsets of $\mathbb{R}^{3}$ are considered. The existence of an invariant measure is proved.

Stochastic approximation properties in Banach spaces

V. P. Fonf, W. B. Johnson, G. Pisier, D. Preiss (2003)

Studia Mathematica

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We show that a Banach space X has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if X has nontrivial type. If for every Radon probability on X, there is an operator from an $L_{p}$ space into X whose range has probability one, then X is a quotient of an $L_{p}$ space. This extends a theorem of Sato’s which dealt with the case p = 2. In any infinite-dimensional Banach space X there is a compact set K...

Relaxation theorem for set-valued functions with decomposable values

Andrzej Kisielewicz (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has $f χ_{A} + g χ_{T} \in K$ . Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.

Maximum principle for forward-backward doubly stochastic control systems and applications

Liangquan Zhang, Yufeng Shi (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short)....

Partially observed optimal controls of forward-backward doubly stochastic systems

Yufeng Shi, Qingfeng Zhu (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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The partially observed optimal control problem is considered for forward-backward doubly stochastic systems with controls entering into the diffusion and the observation. The maximum principle is proven for the partially observable optimal control problems. A probabilistic approach is used, and the adjoint processes are characterized as solutions of related forward-backward doubly stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied...

Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

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Consider a proper metric space and a sequence ${(F ₙ)}_{n \geq 0}$ of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) $X ₙ^{x} = F ₙ \circ . . . \circ F ₁ (x)$ starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying...

Factorization of vector measures and their integration operators

José Rodríguez (2016)

Colloquium Mathematicae

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Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{ν} : L ¹ (ν) \to X$ is also analyzed. As a result, we prove that if $I_{ν}$ is both completely...

Research problems of Jerzy Zabczyk

Szymon Peszat, Łukasz Stettner (2015)

Banach Center Publications

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In the paper we present a selected variety of problems studied by Professor Jerzy Zabczyk. Important part of Prof. Zabczyk's scientific activity was devoted to his PhD students. He has promoted 9 PhD students: Tomasz Bielecki, Jarosław Sobczyk, Łukasz Stettner and Gianmario Tessitore work mostly in control and its applications to mathematical finance, whereas the research of Anna Chojnowska-Michalik, Wojciech Jachimiak, Anna Milian, Szymon Peszat and Anna Rusinek is concentrated mostly...

Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations

Jianhui Huang, Jingtao Shi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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This paper deals with the optimal control problem in which the controlled system is described by a fully coupled anticipated forward-backward stochastic differential delayed equation. The maximum principle for this problem is obtained under the assumption that the diffusion coefficient does not contain the control variables and the control domain is not necessarily convex. Both the necessary and sufficient conditions of optimality are proved. As illustrating examples, two kinds of linear...

Maximum principle for forward-backward doubly stochastic control systems and applications

Liangquan Zhang, Yufeng Shi (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

Stochastic diffrential equations on Banach spaces and their optimal feedback control

(2012)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C₀-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control...

Displaying similar documents to “Topological dual of B ∞ ( I , ₁ ( X , Y ) ) with application to stochastic systems on Hilbert space”

Displaying similar documents to “Topological dual of $B_{\infty} (I, ₁ (X, Y))$ with application to stochastic systems on Hilbert space”