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 Doob–Meyer 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 Euler–Maruyama approximation is used mainly to prove the uniqueness of martingale problems and the smoothness of solutions of stochastic differential equations.

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Kusuoka, Shigeo

0009172

Springer Nature Singapore Pte Ltd Fka Springer Science + Business Media Singapore Pte Ltd

Monographs in Mathematical Economics, 3, 1st ed. 2020, Singapore, 2020

Monographs in mathematical economics, volume 3, Singapore, 2020

Springer Nature, Singapore, 2020

Singapore, Singapore

sm84052500

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Preface 7
Contents 8
Notation 10
1 Preparations from Probability Theory 12
1.1 Review 12
1.2 Lp-Space 19
1.3 Conditional Expectation 22
1.4 Jensen's Inequality for Conditional Expectations 28
1.5 Some Remarks 30
2 Martingale with Discrete Parameter 32
2.1 Definition of Martingale 32
2.2 Doob's Inequality 33
2.3 Stopping Time 35
2.4 Doob's Decomposition and Martingale Transformation 37
2.5 Upcrossing Number and Downcrossing Number 42
2.6 Uniform Integrability 46
2.7 Lévy's Theorem 50
2.8 A Remark on Doob's Decomposition 51
3 Martingale with Continuous Parameter 54
3.1 Several Notions on Stochastic Processes 54
3.2 D-Modification 56
3.3 Doob's Inequality and Stopping Times 59
3.4 Doob–Meyer Decomposition 64
3.5 Quadratic Variation 72
3.6 Continuous Local Martingale 77
3.7 Brownian Motion 84
3.8 Optimal Stopping Time 90
4 Stochastic Integral 97
4.1 Spaces of Stochastic Processes 97
4.2 Continuous Semi-martingales 105
4.3 Itô's Formula 110
5 Applications of Stochastic Integral 115
5.1 Characterization of Brownian Motion 115
5.2 Representation of Continuous Local Martingale 120
5.3 Girsanov Transformation 122
5.4 Moment Inequalities 128
5.5 Itô's Representation Theorem 131
5.6 Property of Brownian Motion 137
5.7 Tanaka's Formula 142
6 Stochastic Differential Equation 145
6.1 Itô's Stochastic Differential Equation and Euler–Maruyama Approximation 145
6.2 Definition of Stochastic Differential Equation 158
6.3 Uniqueness of a Solution to Martingale Problem 166
6.4 Time Homogeneous Stochastic Differential Equation 169
6.5 Smoothness of Solutions to Stochastic Differential Equations 173
7 Application to Finance 188
7.1 Basic Situation and Dynamical Portfolio Strategy 188
7.2 Black–Scholes Model 192
7.3 General Case 198
7.4 American Derivative 208
8 Appendices 211
8.1 Dynkin's Lemma 211
8.2 Convex Function 212
8.3 L2-Weakly Compact 214
8.4 Generalized Inverse Matrix 216
8.5 Proof of Theorem 5.6.1 217
8.6 Gronwall's Inequality 222
Appendix References 223
223
Index 224

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

Chapter 1. Preparations from probability theory -- Chapter 2. Martingale with discrete parameter -- Chapter 3. Martingale with continuous parameter -- Chapter 4. Stochastic integral -- Chapter 5. Applications of stochastic integral -- Chapter 6. Stochastic differential equation -- Chapter 7. Application to finance -- Chapter 8. Appendices -- References

Monographs in Mathematical Economics
Erscheinungsdatum: 20.10.2020

2020-10-27