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Book Reviews – Stochastic Processes and Calculus

Book Reviews - Stochastic Processes and Calculus

This textbook gives a comprehensive introduction to stochastic processes and calculus in the fields of finance and economics, more specifically mathematical finance and time series econometrics. Over the past decades stochastic calculus and processes have gained great importance, because they play a decisive role in the modeling of financial markets and as a basis for modern time series econometrics. Mathematical theory is applied to solve stochastic differential equations and to derive limiting results for statistical inference on nonstationary processes.

This introduction is elementary and rigorous at the same time. On the one hand it gives a basic and illustrative presentation of the relevant topics without using many technical derivations. On the other hand many of the procedures are presented at a technically advanced level: for a thorough understanding, they are to be proven. In order to meet both requirements jointly, the present book is equipped with a lot of challenging problems at the end of each chapter as well as with the corresponding detailed solutions. Thus the virtual text – augmented with more than 60 basic examples and 40 illustrative figures – is rather easy to read while a part of the technical arguments is transferred to the exercise problems and their solutions.

1 Introduction . . . . . . . . . . . . . . . . . . . . . 1
1.1 Summary.. . . . . . . . . . . . . . . . . . . . . . 1
1.2 Finance. . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Econometrics . . . . . . . . . . . . . . . . . . . . . 3
1.4 Mathematics . . . . .  . . . . . . . . . . . . . . . . 6
1.5 Problems and Solutions . . . . . . . . . . . 7
Part I Time Series Modeling
2 Basic Concepts from Probability Theory . . . . 13
2.1 Summary.. . . . . .  . . . . . . . . . . . . . . . . . 13
2.2 Random Variables . . . . . . . . . . . . . . . . . . . . 13
2.3 Joint and Conditional Distributions . . . . . 22
2.4 Stochastic Processes (SP) . . . . . . . . . . 29
2.5 Problems and Solutions . . . . . . . . . . . . . . . . 35
3 AutoregressiveMoving Average Processes (ARMA). . . 45
3.1 Summary. . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Moving Average Processes . . . . . . . . . . 45
3.3 Lag Polynomials and Invertibility . . . . . 51
3.4 Autoregressive and Mixed Processes . 56
3.5 Problems and Solutions . . . . . . . . . . . 68
4 Spectra of Stationary Processes . . . . . . . . . . . . 77
4.1 Summary.. . . .  . . . . . . . . . . . . . . . . . 77
4.2 Definition and Interpretation . . . . . . . . 77
4.3 Filtered Processes . . . . . . . . . . . . . . . . . . . . 84
4.4 Examples of ARMA Spectra . . . . . 89
4.5 Problems and Solutions . . . . . . . . . . 95
5 Long Memory and Fractional Integration 103
5.1 Summary. . . . . . . . . . . . . . . . . . . 103
5.2 Persistence and Long Memory . . . . 103
5.3 Fractionally Integrated Noise . . . . . . . 108
5.4 Generalizations . . . . . . . . . . . . . . . . . . 113
5.5 Problems and Solutions . . . . . . . . . 118
6 Processes with Autoregressive Conditional Heteroskedasticity (ARCH) 127
6.1 Summary.. . . . . . . . . . . . . . . . . . . . 127
6.2 Time-Dependent Heteroskedasticity 127
6.3 ARCH Models . . . . . . . . . . . . . . . 130
6.4 Generalizations . . . . . . . . . . . . . . . . . . 135
6.5 Problems and Solutions . . . . . . . . . . 142
Part II Stochastic Integrals
7 Wiener Processes (WP) . . . . . . . . . . . . . . . 151
7.1 Summary . . . . . . . . . . . . . . . . . . . . . 151
7.2 From RandomWalk to Wiener Process . . . . 151
7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.4 Functions of Wiener Processes . . . . . . 161
7.5 Problems and Solutions . . . . . . . . . . . . . 170
8 Riemann Integrals . . . . . . . . . . . . . . . . . . . . . . 179
8.1 Summary . . . . . . . . . . . . . . . . . . . . 179
8.2 Definition and Fubini’s Theorem . . . 179
8.3 Riemann Integration ofWiener Processes . . . . 183
8.4 Convergence in Mean Square . . . . . . . 186
8.5 Problems and Solutions . . . . . . . . . . . 190
9 Stieltjes Integrals . . . . . . . . . . . . . . . . . . . . . 199
9.1 Summary . . . . . . . . . . . . . . . . . . . . 199
9.2 Definition and Partial Integration . . . . . 199
9.3 Gaussian Distribution and Autocovariances . . . 202
9.4 Standard Ornstein-Uhlenbeck Process . . 204
9.5 Problems and Solutions . . . . . . . . . . . 207
10 Ito Integrals . . . . . . . . . . . . . . . . . . . . . . 213
10.1 Summary . . . . . . . . . . . . . . . . 213
10.2 A Special Case . . . . . . . . . . . . . . . . . . 213
10.3 General Ito Integrals . . . . . . . . . . . . 218
10.4 (Quadratic) Variation . . . . . . . . . . . . 222
10.5 Problems and Solutions . . . . . . . . . . 229
11 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 239
11.1 Summary . . . . . . . . . . . . . . . . . . . 239
11.2 The Univariate Case . . . . . . . . . . . . . . . 239
11.3 Bivariate Diffusions with One WP . . . . 245
11.4 Generalization for IndependentWP . . 250
11.5 Problems and Solutions . . . . . . . . . . . 254
Part III Applications
12 Stochastic Differential Equations (SDE) . . . . 261
12.1 Summary . . . . . . . . . . . . . . . . . . . 261
12.2 Definition and Existence . . . . . . . . . . . 261
12.3 Linear Stochastic Differential Equations 265
12.4 Numerical Solutions . . . . . . . . . . . . . . . . 272
12.5 Problems and Solutions . . . . . . . . . . 273
13 Interest Rate Models . . . . . . . . . . . . . . . . . . 285
13.1 Summary . . . . . . . . . . . . . . . . . . . . 285
13.2 Ornstein-Uhlenbeck Process (OUP) . . 285
13.3 Positive Linear Interest Rate Models . . . 288
13.4 Nonlinear Models . . . . . . . . . . . . . . . 292
13.5 Problems and Solutions . . . . . . . . . . . 296
14 Asymptotics of Integrated Processes . . . . . . . . . 303
14.1 Summary . . . . . . . . . . . . . . . . . . . . . 303
14.2 Limiting Distributions of Integrated Processes 303
14.3 Weak Convergence of Functions . . . . 310
14.4 Multivariate Limit Theory . . . . . . . . . . . . . 317
14.5 Problems and Solutions . . . . . . . . . . . . . . . 321
15 Trends, Integration Tests and Nonsense Regressions 331
15.1 Summary . . . . . . . . . . . . . . . . . . 331
15.2 Trend Regressions . . . . . . . . . . . . . 331
15.3 Integration Tests . . . . . . . . . . . . . . . . 336
15.4 Nonsense Regression . . . . . . . . . . . . . . . 341
15.5 Problems and Solutions . . . . . . . . . . 344
16 Cointegration Analysis . . . . . . . . . . . . . . . . . 353
16.1 Summary . . . . . . . . . . . . . . . . 353
16.2 Error-Correction and Cointegration . . 353
16.3 Cointegration Regressions . . . . . . . . 358
16.4 Cointegration Testing . . . . . . . . . . . 365
16.5 Problems and Solutions . . . . . . . . . 373

Book Reviews – Stochastic Processes and Calculus


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