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Documents  93E24 | enregistrements trouvés : 4

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Statistical learning theory and stochastic optimization :école d'été de probabilités de Saint-Flour XXXI, 2001 Catoni, Olivier ; Picard, Jean | Springer 2004

Congrès

- 272 p.
ISBN 978-3-540-22572-0

Lecture notes in mathematics , 1851

Localisation : Collection 1er étage

probabilité # optimisation # optimisation stochastique # chaîne de Markov # contrôle stochastique # apprentissage statistique # estimation de Gibbs # classification adaptative # regression des moindres carrés # divergence de Kullback # estimateur # codage # reconnaissance de forme

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Least squares regression Monte Carlo for approximating BSDES and semilinear PDES Turkedjiev, Plamen | CIRM H

Multi angle

Research schools;Control Theory and Optimization;Probability and Statistics

In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the of the continuous time solution of the BSDE. The nodes of the discrete time processes can be expressed as conditional expectations. As we shall demonstrate, the choice of discrete time process is very important, as its properties will impact the performance of the overall numerical scheme. In the second stage, the conditional expectation is approximated in functional form using least squares regression on synthetically generated data - Monte Carlo simulations drawn from a suitable probability distribution. A key feature of the regression step is that the explanatory variables are built on a user chosen finite dimensional linear space of functions, which the user specifies by setting basis functions. The choice of basis functions is made on the hypothesis that it contains the solution, so regularity and boundedness assumptions are used in its construction. The impact of the choice of the basis functions is exposed in error estimates.
In addition to the choice of discrete time approximation and the basis functions, the Markovian structure of the problem gives significant additional freedom with regards to the Monte Carlo simulations. We demonstrate how to use this additional freedom to develop generic stratified sampling approaches that are independent of the underlying transition density function. Moreover, we demonstrate how to leverage the stratification method to develop a HPC algorithm for implementation on GPUs.
Thanks to the Feynmann-Kac relation between the the solution of a BSDE and its associated semilinear PDE, the approximation of the BSDE can be directly used to approximate the solution of the PDE. Moreover, the smoothness properties of the PDE play a crucial role in the selection of the hypothesis space of regressions functions, so this relationship is vitally important for the numerical scheme.
We conclude with some draw backs of the regression approach, notably the curse of dimensionality.
In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the ...

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Handbook of Partial Least Squares :Concepts, Methods and Applications Esposito Vinzi, Vincenzo ; Chin, Wynne W. ; Henseler, Jörg ; Wang, Huiwen | Springer 2010

Ouvrage

- XIV-798 p.
ISBN 978-3-540-32825-4

Springer Handbooks of Computational Statistics

moindres carrés # analyse de régression # modèle d'équations structurales # analyse de parcours # informatique # variable latente # marketing # méthode statistique

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Lectures on Weiner and Kalman filtering Kailath, Thomas | Springer-Verlag 1981

Ouvrage

- 187 p.
ISBN 978-3-211-81664-6

CISM courses and lectures , 140

Localisation : Ouvrage RdC (KAIL)

application des processus au filtrage du signal # commande # filtrage de Kalman # filtrage de Weiner # méthode des moindres carrés # système stochastique et contrôle # théorie des probabilités et processus stochastiques # théorie des systèmes

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