简介
The theory of stochastic integration, also called the Ito calculus, has a large spectrum of applications in virtually every scientific area involving random functions, but it can be a very difficult subject for people without much mathematical background. The Ito calculus was originally motivated by the construction of Markov diffusion processes from infinitesimal generators. Previously, the construction of such processes required several steps, whereas Ito constructed these diffusion processes directly in a single step as the solutions of stochastic integral equations associated with the infinitesimal generators. Moreover, the properties of these diffusion processes can be derived from the stochastic integral equations and the Ito formula. This introductory textbook on stochastic integration provides a concise introduction to the Ito calculus, and covers the following topics: * Constructions of Brownian motion; * Stochastic integrals for Brownian motion and martingales; * The Ito formula; * Multiple Wiener-Ito integrals; * Stochastic differential equations; * Applications to finance, filtering theory, and electric circuits. The reader should have a background in advanced calculus and elementary probability theory, as well as a basic knowledge of measure theory and Hilbert spaces. Each chapter ends with a variety of exercises designed to help the reader further understand the material. Hui-Hsiung Kuo is the Nicholson Professor of Mathematics at Louisiana State University. He has delivered lectures on stochastic integration at Louisiana State University, Cheng Kung University, Meijo University, and University of Rome "Tor Vergata," among others. He is also the author of Gaussian Measures in Banach Spaces (Springer 1975), and White Noise Distribution Theory (CRC Press 1996), and a memoir of his childhood growing up in Taiwan, An Arrow Shot into the Sun (Abridge Books 2004).
目录
12 Rand Walks .............. . ... ....... ... 1
Rsandom Waks ................. 4...... 6
Exercises ........ .. . .. ... ... ... .... ... .. . ...... .......... 6
2 Brownian Motion . . .. . ...... .... ...... ... 7
2.1 Definition of Brownian Motion ... .. ......
2.2 Simple Properties of Brownian Motion ... ..... ...... 8
2 3 Wiener Integral . .. . ...... ..... ............. 9
2.4 Conditional Expectation ..... .... ... ..... . 14
2,5 Martingales ............................. 17
2 6 Series Expansion of Wiener Integrals .... .... ........ .. 20
Exercises - . . . . .... . . .. . ...... . . . . . ... . 21
3 Constructions of Brownian Motion . .. ........ ...... . 23
3 1 W iener Space ........................ .... .... . 23
3,2 Borel-Cantelli Lemma and Chebyshev Inequality ... ... . 25
3.3 Kolmogorov's Extension and Continuity Theorems ....... 27
3.4 Levy's Interpolation M ethod ......................... .. 34
E xercises ... ..... ........... .......................... 35
4 Stochastic Integrals ... ..... .............. .. 37
4.1 Background and Motivation ........................ 37
4.2 Filtrations for a Brownian Motion .............. ....... 41
4.3 Stochastic Integrals ........... ........ 43
4.4 Simple Examples of Stochastic Integrals ..... ... .. .. 48
4.5 Doob Sibimartingale Inequality . ...... ... ........ 51
4 6 Stochastic Processes Defined by Ito Integrals ............. 52
4.7 Riemann Sums and Stochastic Integrals ....... ......... . 57
Exercises . .. . . . . .. . .. . ...... . .. . . ... .... .. . 58
5 An Extension of Stochastic Integrals ........ .. .. ... . 61
5.1 A Larger Class of Integrands ................... 61
5,2 A Key I em m a . ......... . ... .. . .... . ... 64
5.3 General Stochastic Integrals .... ........ ....... . 65
5.4 Stopping Times ............ . .... ...... . . 68
55 Associated Stochastic Processes ................... . 70
E xercises .... . ..... .. .. . . .... .. . ... .. ... . . . . . ... .... ..... . 73
6 Stochastic Integrals for Martingales - ...................7
6.2 Poisson Processes. .. ...... .... .. ... ......... 76
6.3 Piedictable Stocrhastic Proesses ... ... ........ 79
6.4 Doob-Meyer Decomposition 0 heorem ........ ........ 80
65 aingales as Integrators ....... .......... 84
6 6 Extension for Integrands 89
7 T he It6 Form ula .. .. . ... ........................... .... 93
7.1 It's Formula in the Simplest Form . 93
7.2 Proof of to's Formni a ........... .. ... ..... . . ... 96
7.3 1 Io's Formula Slightly Generalized ......................... 99
7.4 Itos Formula in the General Fornm .. . . ... . . ..... 2102
7.5 Multidiensional Ito's Formula . .. 106
7.6 to's Formula for Martingales ..... . ... 109
Exercises .... . . . ...... ........... . . . . . . .. .. .. .. . 113
8 Applications of the Ito Formula . ........ ...... . . 115
8 1 Evaluation of Stochastic integrals ...... 115
8.2 Decorposition and Compensators ... .. ... ..... .... .117
8.3 Stratonovich Integral . .. . .. ... ... ... . .. ... . 119
.4 Levy's Characterization Theorem ...... ......... . 124
8.5 Multidimensional Brownian Motions .o.. . . . .. .. . . 129
8.6 Tanaka's Formula and Local Time ...,.. ..... .. .. . 133
.7 Exponential Processes, . . ...... .. . . .. .. .. . . 36
8.8 Transformation of Probability Measures ............... 138
8.9 Girsanov Theorem .... .. ...... ... .... . . .141
Exercises... . .... 145
9 Multiple Wiener Ito Integrals ........ ............. 1147
9.1 A Simple Example .. ................ .. 147
9.2 Double Wiener-f I Integrals . .......... . .. . 150
9.3 Herrite Polynonials . ....... . .. .. ..... ... ... . 155
9.4 Homogeneous Chaos ................ . ......... 159
9.5 Orthonorinal Basis for Homogeneous Chaos ......... ... 164
96 Multiple Wiener-t I ntegrals ....... ....... . 68
"9.7 W te iener-It6 Tih orem .. . . ... .. . ........ . .... . . ... 176
9.8 Representation of Brownian Martingales ......... .... 180
Exercises . .. . .. ...... .... ...... ........ ... 183
10 Stochastic Differential Equations ....................... 185
10.1 Some Examples .... ..... ............ .... 185
10.2 Bellm an -Gronwall Inequalihty . .. .. .. . ... ........ 188
10.3 Existene and Uniqueness Theorem . ...... ........ ... 190
10.4 Systems of Stochastic Differential Equations .. .......... 196
10.5 Markov Property ....................... . 197
10.6 Solutions of Stochastic Differential Equations ... ...... 203
10.7 Some Estimates for the Solutions .... .. ..... ....... . 208
10.8 Diffusion Processes. ...... ... ... ...... . ....... 211
10.9 Semigroups and the Kolmogorov Equations ..... ..... 216
E xercises . . . . .. . . . . ... .. . . ..... ...... . . ... . 229
11 Some Applications and Additional Topics ....... ..... 231
11 1 Linear Stochastic Differential Equations .............. 231
11.2 Application to Finance ........... ............ . 234
11 3 Application to Filtering Theory ............... ... ... 246
11.4 ninynman Kac Formula, .. . .. ....... ....... . .249
11.5 Approximation of Stochasti Integrals ............... . 254
11 6 White Noise and Electric Circuits ....... ...... 258
Exercises .......... ....... .. ... . ...... 265
- 名称
- 类型
- 大小
光盘服务联系方式: 020-38250260 客服QQ：4006604884
云图客服：
用户发送的提问，这种方式就需要有位在线客服来回答用户的问题，这种 就属于对话式的,问题是这种提问是否需要用户登录才能提问
Video Player
×
Audio Player
×
pdf Player
×
