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Author: Bayer, Christian × Author: Eigel, Martin × End date: 2022 × Version: publishedVersion × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 3 of 3
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    Pricing high-dimensional Bermudan options with hierarchical tensor formats
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bayer, Christian; Eigel, Martin; Sallandt, Leon; Trunschke, Philipp
    An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the ``curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.
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    A fully adaptive interpolated stochastic sampling method for random PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Anker, Felix; Bayer, Christian; Eigel, Martin; Neumann, Johannes; Schoenmakers, John
    A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.
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    SDE based regression for random PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Anker, Felix; Bayer, Christian; Eigel, Martin; Ladkau, Marcel; Neumann, Johannes; Schoenmakers, John G.M.
    A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.