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Asymptotic Expansion and Weak Approximation: Applications of Malliavin Calculus and Deep Learning

By (author) Akihiko Takahashi, By (author) Toshihiro Yamada

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This book provides a self-contained lecture on a Malliavin calculus approach to asymptotic expansion and weak approximation of stochastic differential equations (SDEs), Â along with numerical methods for computing parabolic partial differential equations (PDEs). Constructions of weak approximation and asymptotic expansion are given in detail using Malliavinâ€™s integration by parts with theoretical convergence analysis. Weak approximation algorithms and Python codes are available with numerical examples. Moreover, the weak approximation scheme is effectively applied to high-dimensional nonlinear problems without suffering from the curse of dimensionality through combining with a deep learning method. Readers including graduate-level students, researchers, and practitioners can understand both theoretical and applied aspects of recent developments of asymptotic expansion and weak approximation.

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The Coming Wave: AI, Power and Our Future

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The Order of Time

Carlo Rovelli
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224 pages
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Originally published in Italian: L'ordine del tempo (Milan: Adelphi Edizioni, 2017).

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