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- ItemSolving linear parabolic rough partial differential equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bayer, Christian; Belomestny, Denis; Redmann, Martin; Riedel, Sebastian; Schoenmakers, JohnWe study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity with 1=3 < 1=2. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.
- ItemRegression on particle systems connected to mean-field SDEs with applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Belomestny, Denis; Schoenmakers, John G.M.In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.