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Equivalent cost functionals and stochastic linear quadratic optimal control problems

Zhiyong Yu (2013)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the stochastic linear quadratic optimal control problems (LQ problems, for short) for which the coefficients are allowed to be random and the cost functionals are allowed to have negative weights on the square of control variables. We propose a new method, the equivalent cost functional method, to deal with the LQ problems. Comparing to the classical methods, the new method is simple, flexible and non-abstract. The new method can also be applied to deal with nonlinear...

Ergodic behaviour of stochastic parabolic equations

Jan Seidler (1997)

Czechoslovak Mathematical Journal

The ergodic behaviour of homogeneous strong Feller irreducible Markov processes in Banach spaces is studied; in particular, existence and uniqueness of finite and σ -finite invariant measures are considered. The results obtained are applied to solutions of stochastic parabolic equations.

Ergodic control of linear stochastic equations in a Hilbert space with fractional Brownian motion

Tyrone E. Duncan, B. Maslowski, B. Pasik-Duncan (2015)

Banach Center Publications

A linear-quadratic control problem with an infinite time horizon for some infinite dimensional controlled stochastic differential equations driven by a fractional Brownian motion is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well known linear feedback control for the associated infinite dimensional deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system...

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

Ľubomír Baňas, Zdzisław Brzeźniak, Mikhail Neklyudov, Martin Ondreját, Andreas Prohl (2015)

Czechoslovak Mathematical Journal

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also...

Ergodicity of hypoelliptic SDEs driven by fractional brownian motion

M. Hairer, N. S. Pillai (2011)

Annales de l'I.H.P. Probabilités et statistiques

We demonstrate that stochastic differential equations (SDEs) driven by fractional brownian motion with Hurst parameter H>½ have similar ergodic properties as SDEs driven by standard brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the...

Espaces de Sobolev gaussiens

Denis Feyel, A. de La Pradelle (1989)

Annales de l'institut Fourier

Soit μ une mesure gaussienne sur un espace localement convexe E . On donne un nouveau point de vue sur le premier espace de Sobolev W ( E , μ ) construit sur E et μ . La différentielle f ' de f ∈ W ( E , μ ) est une fonction de deux variables ( x , y ) ∈ E × E , “quasi-linéaire” dans la seconde variable.La différentielle d’une intégrale stochastique est une intégrale stochastique sur E × E muni de μ × μ .On montre que la “procapacité gaussienne” naturelle est une vraie capacité si E est un espace de Banach ou de Fréchet ou le dual faible d’un espace...

Essential m-dissipativity of Kolmogorov operators corresponding to periodic 2 ⁢ D -Navier Stokes equations

Viorel Barbu, Giuseppe Da Prato, Arnaud Debussche (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove the essential m-dissipativity of the Kolmogorov operator associated with the stochastic Navier-Stokes flow with periodic boundary conditions in a space L 2 ⁢ H , ν where ν is an invariant measure

Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations.

Vassili N. Kolokol'tsov, René L. Schilling, Alexei E. Tyukov (2004)

Revista Matemática Iberoamericana

We study stochastic Hamilton-Jacobi-Bellman equations and the corresponding Hamiltonian systems driven by jump-type Lévy processes. The main objective of the present papel is to show existence, uniqueness and a (locally in time) diffeomorphism property of the solution: the solution trajectory of the system is a diffeomorphism as a function of the initial momentum. This result enables us to implement a stochastic version of the classical method of characteristics for the Hamilton-Jacobi equations....

Estimation in models driven by fractional brownian motion

Corinne Berzin, José R. León (2008)

Annales de l'I.H.P. Probabilités et statistiques

Let {bH(t), t∈ℝ} be the fractional brownian motion with parameter 0<H<1. When 1/2<H, we consider diffusion equations of the type X(t)=c+∫0tσ(X(u)) dbH(u)+∫0tμ(X(u)) du. In different particular models where σ(x)=σ or σ(x)=σ  x and μ(x)=μ or μ(x)=μ  x, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(⋅)...

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