This is an introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. Based on measure theory, which is introduced as smoothly as possible, it provides practical skills in the use of MAPLE in the context of probability and its applications. It offers to graduates and advanced undergraduates an overview and intuitive background for more advanced studies.
Author(s): Sasha Cyganowski, Peter Kloeden, Jerzy Ombach
Series: Universitext
Edition: 1
Publisher: Springer
Year: 2001
Language: English
Pages: 324
Preface......Page 6
Notation......Page 8
Maple......Page 10
Table of Contents......Page 14
1.1 The Definition of Probability......Page 18
1.2 The Classical Scheme and Its Extensions......Page 23
1.3 Geometric Probability......Page 31
1.4 Conditional and Total Probability......Page 33
1.5 Independent Events......Page 39
1.6 MAPLE Session......Page 44
1.7 Exercises......Page 46
2.1 Measure......Page 49
2.2 Integral......Page 54
2.3 Properties of the Integral......Page 63
2.4 Determining Integrals......Page 67
2.5 MAPLE Session......Page 68
2.6 Exercises......Page 75
3 Random Variables and Distributions......Page 77
3.1 Probability Distributions......Page 79
3.2 Random Variables and Random Vectors......Page 85
3.3 Independence......Page 88
3.4 Functions of Random Variables and Vectors......Page 90
3.5 MAPLE Session......Page 94
3.6 Exercises......Page 98
4.1 Mathematical Expectation......Page 101
4.2 Variance......Page 105
4.3 Computation of Moments......Page 108
4.4 Chebyshev's Inequality......Page 114
4.5 Law of Large Numbers......Page 116
4.6 Correlation......Page 123
4.7 MAPLE Session......Page 128
4.8 Exercises......Page 133
5.1 Counting......Page 136
5.2 Waiting Times......Page 143
5.3 The Normal Distribution......Page 151
5.4 Central Limit Theorem......Page 155
5.5 Multidimensional Normal Distribution......Page 162
5.6 MAPLE Session......Page 168
5.7 Exercises......Page 173
6.1 Pseudo-Random Number Generation......Page 175
6.2 Basic Statistical Tests......Page 179
6.3 The Runs Test......Page 182
6.4 Goodness of Fit Tests......Page 188
6.5 Independence Test......Page 191
6.6 Confidence Intervals......Page 193
6.7 Inference for Numerical Simulations......Page 196
6.8 MAPLE Session......Page 199
6.9 Exercises......Page 204
7.1 Conditional Expectation......Page 206
7.2 Markov Chains......Page 211
7.3 Special Classes of Stochastic Processes......Page 223
7.4 Continuous-Time Stochastic Processes......Page 224
7.5 Continuity and Convergence......Page 231
7.6 MAPLE Session......Page 233
7.7 Exercises......Page 239
8.1 Introduction......Page 241
8.2 Ito Stochastic Integrals......Page 242
8.3 Stochastic Differential Equations......Page 245
8.4 Stochastic Chain Rule: the Ito Formula......Page 248
8.5 Stochastic Taylor Expansions......Page 250
8.6 Stratonovich Stochastic Calculus......Page 253
8.7 MAPLE Session......Page 256
8.8 Exercises......Page 258
9.1 Solving Scalar Stratonovich SDEs......Page 260
9.2 Linear Scalar SDEs......Page 264
9.3 Scalar SDEs Reducible to Linear SDEs......Page 268
9.4 Vector SDES......Page 271
9.5 Vector Linear SDE......Page 276
9.6 Vector Stratonovich SDEs......Page 279
9.7 MAPLE Session......Page 281
9.8 Exercises......Page 286
10.1 Numerical Methods for ODEs......Page 288
10.2 The Stochastic Euler Scheme......Page 291
10.3 How to Derive Higher Order Schemes......Page 295
10.4 Higher Order Strong Schemes......Page 299
10.5 Higher Order Weak Schemes......Page 301
10.6 The Euler and Milstein Schemes for Vector SDEs......Page 302
10.7 MAPLE Session......Page 306
10.8 Exercises......Page 312
Bibliographical Notes......Page 314
References......Page 316
Index......Page 318
