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About the author
Yuliya Mishura, National University of Kyiv, Ukraine Georgiy Shevchenko, National University of Kyiv, UkraineThis book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with contemporary subjects: integration with respect to Gaussian processes, Itȏ integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models.
400 pages; ISBN 9781119476597
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Title: Theory and Statistical Applications of Stochastic Processes
Author: Yuliya Mishura; Georgiy Shevchenko
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 Contents
Preface xi
Introduction xiii
Part 1 Theory of Stochastic Processes 1
Chapter 1 Stochastic Processes General Properties. Trajectories, Finitedimensional Distributions 3
1.1 Definition of a stochastic process 3
1.2 Trajectories of a stochastic process Some examples of stochastic processes 5
1.2.1 Definition of trajectory and some examples 5
1.2.2 Trajectory of a stochastic process as a random element.8
1.3 Finitedimensional distributions of stochastic processes: consistency conditions.10
1.3.1 Definition and properties of finitedimensional distributions 10
1.3.2 Consistency conditions.11
1.3.3 Cylinder sets and generated σalgebra 13
1.3.4 Kolmogorov theorem on the construction of a stochastic process by the family of probability distributions 15
1.4 Properties of σalgebra generated by cylinder sets. The notion of σalgebra generated by a stochastic process 19
Chapter 2 Stochastic Processes with Independent Increments 21
2.1 Existence of processes with independent increments in terms of incremental characteristic functions 21
2.2 Wiener process 24
2.2.1 Onedimensional Wiener process 24
2.2.2 Independent stochastic processes Multidimensional Wiener process 24
2.3 Poisson process 27
2.3.1 Poisson process defined via the existence theorem 27
2.3.2 Poisson process defined via the distributions of the increments 28
2.3.3 Poisson process as a renewal process 30
2.4 Compound Poisson process 33
2.5 Lévy processes 34
2.5.1 Wiener process with a drift 36
2.5.2 Compound Poisson process as a Lévy process 36
2.5.3 Sum of a Wiener process with a drift and a Poisson process 36
2.5.4 Gamma process 37
2.5.5 Stable Lévy motion37
2.5.6 Stable Lévy subordinator with stability parameter α ∈ (0, 1) 38
Chapter 3 Gaussian Processes Integration with Respect to Gaussian Processes 39
3.1 Gaussian vectors 39
3.2 Theorem of Gaussian representation (theorem on normal correlation) 42
3.3 Gaussian processes. 44
3.4 Examples of Gaussian processes 46
3.4.1 Wiener process as an example of a Gaussian process 46
3.4.2 Fractional Brownian motion.48
3.4.3 Subfractional and bifractional Brownian motion 50
3.4.4 Brownian bridge 50
3.4.5 Ornstein–Uhlenbeck process 51
3.5 Integration of nonrandom functions with respect to Gaussian processes 52
3.5.1 General approach 52
3.5.2 Integration of nonrandom functions with respect to the Wiener process 54
3.5.3 Integration w.r.t the fractional Brownian motion 57
3.6 Twosided Wiener process and fractional Brownian motion: Mandelbrot–van Ness representation of fractional Brownian motion 60
3.7 Representation of fractional Brownian motion as the Wiener integral on the compact integral 63
Chapter 4 Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion 67
4.1 Construction of a Wiener process on the interval [0, 1] 67
4.2 Construction of a Wiener process on R+ 72
4.3 Nowhere differentiability of the trajectories of a Wiener process 74
4.4 Power variation of the Wiener process and of the fractional Brownian motion77
4.4.1 Ergodic theorem for power variations 77
4.5 Selfsimilar stochastic processes 79
4.5.1 Definition of selfsimilarity and some examples 79
4.5.2 Power variations of selfsimilar processes on finite intervals.80
Chapter 5 Martingales and Related Processes 85
5.1 Notion of stochastic basis with filtration 85
5.2 Notion of (sub, super) martingale: elementary properties 86
5.3 Examples of (sub, super) martingales 87
5.4 Markov moments and stopping times 90
5.5 Martingales and related processes with discrete time 96
5.5.1 Upcrossings of the interval and existence of the limit of submartingale 96
5.5.2 Examples of martingales having a limit and of uniformly and nonuniformly integrable martingales 102
5.5.3 Lévy convergence theorem 104
5.5.4 Optional stopping 105
5.5.5 Maximal inequalities for (sub, super) martingales 108
5.5.6 Doob decomposition for the integrable processes with discrete time 111
5.5.7 Quadratic variation and quadratic characteristics: Burkholder–Davis–Gundy inequalities 113
5.5.8 Change of probability measure and Girsanov theorem for discretetime processes 116
5.5.9 Strong law of large numbers for martingales with discrete time 120
5.6 Lévy martingale stopped 126
5.7 Martingales with continuous time 127
Chapter 6 Regularity of Trajectories of Stochastic Processes 131
6.1 Continuity in probability and in L2(Ω,F, P) 131
6.2 Modification of stochastic processes: stochastically equivalent and indistinguishable processes 133
6.3 Separable stochastic processes: existence of separable modification 135
6.4 Conditions of Dregularity and absence of the discontinuities of the second kind for stochastic processes 138
6.4.1 Skorokhod conditions of Dregularity in terms of threedimensional distributions 138
6.4.2 Conditions of absence of the discontinuities of the second kind formulated in terms of conditional probabilities of large increments 144
6.5 Conditions of continuity of trajectories of stochastic processes 148
6.5.1 Kolmogorov conditions of continuity in terms of twodimensional distributions 148
6.5.2 Hölder continuity of stochastic processes: a sufficient condition 152
6.5.3 Conditions of continuity in terms of conditional probabilities 154
Chapter 7 Markov and Diffusion Processes 157
7.1 Markov property 157
7.2 Examples of Markov processes 163
7.2.1 Discretetime Markov chain 163
7.2.2 Continuoustime Markov chain 165
7.2.3 Process with independent increments 168
7.3 Semigroup resolvent operator and generator related to the homogeneous Markov process 168
7.3.1 Semigroup related to Markov process 168
7.3.2 Resolvent operator and resolvent equation 169
7.3.3 Generator of a semigroup.171
7.4 Definition and basic properties of diffusion process 175
7.5 Homogeneous diffusion process Wiener process as a diffusion process 179
7.6 Kolmogorov equations for diffusions 181
Chapter 8 Stochastic Integration 187
8.1 Motivation..187
8.2 Definition of Itô integral 189
8.2.1 Itô integral of Wiener process 195
8.3 Continuity of Itô integral 197
8.4 Extended Itô integral 199
8.5 Itô processes and Itô formula 203
8.6 Multivariate stochastic calculus 212
8.7 Maximal inequalities for Itô martingales 215
8.7.1 Strong law of large numbers for Itô local martingales 218
8.8 Lévy martingale characterization of Wiener process 220
8.9 Girsanov theorem 223
8.10 Itô representation 228
Chapter 9 Stochastic Differential Equations.233
9.1 Definition, solvability conditions, examples 233
9.1.1 Existence and uniqueness of solution 234
9.1.2 Some special stochastic differential equations 238
9.2 Properties of solutions to stochastic differential equations 241
9.3 Continuous dependence of solutions on coefficients 245
9.4 Weak solutions to stochastic differential equations. 247
9.5 Solutions to SDEs as diffusion processe 249
9.6 Viability, comparison and positivity of solutions to stochastic differential equations 252
9.6.1 Comparison theorem for onedimensional projections of stochastic differential equations 257
9.6.2 Nonnegativity of solutions to stochastic differential equations 258
9.7 Feynman–Kac formula 258
9.8 Diffusion model of financial markets 260
9.8.1 Admissible portfolios, arbitrage and equivalent martingale measure 263
9.8.2 Contingent claims, pricing and hedging 266
Part 2 Statistics of Stochastic Processes 271
Chapter 10 Parameter Estimation 273
10.1 Drift and diffusion parameter estimation in the linear regression model with discrete time 273
10.1.1 Drift estimation in the linear regression model with discrete time in the case when the initial value is known 274
10.1.2 Drift estimation in the case when the initial value is unknown 277
10.2 Estimation of the diffusion coefficient in a linear regression model with discrete time 277
10.3 Drift and diffusion parameter estimation in the linear model with continuous time and the Wiener noise 278
10.3.1 Drift parameter estimation 279
10.3.2 Diffusion parameter estimation 280
10.4 Parameter estimation in linear models with fractional Brownian motion 281
10.4.1 Estimation of Hurst index 281
10.4.2 Estimation of the diffusion parameter 283
10.5 Drift parameter estimation 284
10.6 Drift parameter estimation in the simplest autoregressive model 285
10.7 Drift parameters estimation in the homogeneous diffusion model 289
Chapter 11 Filtering Problem KalmanBucy Filter 293
11.1 General setting 293
11.2 Auxiliary properties of the nonobservable process 294
11.3 What is an optimal filter 295
11.4 Representation of an optimal filter via an integral equation with respect to an observable process 296
11.5 Integral WienerHopf equation 299
Appendices 311
Appendix 1 313
Appendix 2 329
Bibliography 363
Index 369
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