Titlebook: Stochastic Analysis; Shigeo Kusuoka Book 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature

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Book 2020grable submartingales are considered, and only elementary facts ofthe square integrable functions are used in the proof. In stochastic differential equations, the Euler–Maruyama approximation is used mainly to prove the uniqueness of martingale problems and the smoothness of solutions of stochastic

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Martingale with Discrete Parameter,Let . be an ordered set. We only consider the case that . is a subset of . in this book. In almost all cases, . is . . or . . or [0,?.],? . or . although we consider the case that . is .

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Applications of Stochastic Integral,We assume that . is a standard filtered probability space throughout this section.

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