β Scribed by Shigeo Kusuoka
No coin nor oath required. For personal study only.
This book is intended for university seniors and graduate students majoring in probability theory or mathematical finance. In the first chapter, results in probability theory are reviewed. Then, it follows a discussion of discrete-time martingales, continuous time square integrable martingales (particularly, continuous martingales of continuous paths), stochastic integrations with respect to continuous local martingales, and stochastic differential equations driven by Brownian motions. In the final chapter, applications to mathematical finance are given. The preliminary knowledge needed by the reader is linear algebra and measure theory. Rigorous proofs are provided for theorems, propositions, and lemmas.
In this book, the definition of conditional expectations is slightly different than what is usually found in other textbooks. For the DoobβMeyer decomposition theorem, only square integrable submartingales are considered, and only elementary facts of the square integrable functions are used in the proof. In stochastic differential equations, the EulerβMaruyama approximation is used mainly to prove the uniqueness of martingale problems and the smoothness of solutions of stochastic differential equations.
Front Matter ....Pages i-xii
Preparations from Probability Theory (Shigeo Kusuoka)....Pages 1-20
Martingale with Discrete Parameter (Shigeo Kusuoka)....Pages 21-42
Martingale with Continuous Parameter (Shigeo Kusuoka)....Pages 43-85
Stochastic Integral (Shigeo Kusuoka)....Pages 87-104
Applications of Stochastic Integral (Shigeo Kusuoka)....Pages 105-134
Stochastic Differential Equation (Shigeo Kusuoka)....Pages 135-177
Application to Finance (Shigeo Kusuoka)....Pages 179-201
Appendices (Shigeo Kusuoka)....Pages 203-214
Back Matter ....Pages 215-218
Mathematics; Probability Theory and Stochastic Processes; Measure and Integration; Functional Analysis; Applications of Mathematics
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