Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences

Estimation of Stochastic Processes is intended for researchers in the field of econometrics, financial mathematics, statistics or signal processing. This book gives a deep understanding of spectral theory and estimation techniques for stochastic processes with stationary increments. It focuses on the estimation of functionals of unobserved values for stochastic processes with stationary increments, including ARIMA processes, seasonal time series and a class of cointegrated sequences.

Furthermore, this book presents solutions to extrapolation (forecast), interpolation (missed values estimation) and filtering (smoothing) problems based on observations with and without noise, in discrete and continuous time domains. Extending the classical approach applied when the spectral densities of the processes are known, the minimax method of estimation is developed for a case where the spectral information is incomplete and the relations that determine the least favorable spectral densities for the optimal estimations are found.

Notations ix

Introduction xi

Chapter 1. Stationary Increments of Discrete Time Stochastic Processes: Spectral Representation1

Chapter 2. Extrapolation Problem for Stochastic Sequences with Stationary nth Increments 9

2.1. The classical method of extrapolation 9

2.2. Minimax (robust) method of extrapolation 21

2.3. Least favorable spectral density in the class D⁰_f 24

2.4. Least favorable spectral densities which admit factorization in the class D⁰_f 25

2.5. Least favorable spectral density in the class D^u_v 29

2.6. Least favorable spectral density which admits factorization in the class D^u_v 29

Chapter 3. Interpolation Problem for Stochastic Sequences with Stationary nth Increments31

3.1. The classical method of interpolation 31

3.2. Minimax method of interpolation 41

3.3. Least favorable spectral densities in the class D⁻_0,n 43

3.4. Least favorable spectral densities in the class D⁻_M,n 47

Chapter 4. Extrapolation Problem for Stochastic Sequences with Stationary nth Increments Based on Observations with Stationary Noise53

4.1. The classical method of extrapolation with noise 53

4.2. Extrapolation of cointegrated stochastic sequences 71

4.3. Minimax (robust) method of extrapolation 75

4.4. Least favorable spectral densities in the class D⁰_f × D⁰_g 80

4.5. Least favorable spectral densities which admit factorization in the class D⁰_f × D⁰_g 82

4.6. Least favorable spectral densities in the class D^u_v × D_ε 84

4.7. Least favorable spectral densities which admit factorization in the class D^u_v × D_ε 86

Chapter 5. Interpolation Problem for Stochastic Sequences with Stationary nth Increments Based on Observations with Stationary Noise89

5.1. The classical method of interpolation with noise 89

5.2. Interpolation of cointegrated stochastic sequences 96

5.3. Minimax (robust) method of interpolation 97

5.4. Least favorable spectral densities in the class D⁻_0,f× D⁻_0,g 100

5.5. Least favorable spectral densities in the class D_2ε1× D_1ε2 103

Chapter 6. Filtering Problem of Stochastic Sequences with Stationary nth Increments Based on Observations with Stationary Noise107

6.1. The classical method of filtering 107

6.2. Filtering problem for cointegrated stochastic sequences 119

6.3. Minimax (robust) method of filtering 124

6.4. Least favorable spectral densities in the class D⁰_f × D⁰_g 129

6.5. Least favorable spectral densities which admit factorization in the class D⁰_f × D⁰_g 131

6.6. Least favorable spectral densities in the class D^u_v × D_ε 134

6.7. Least favorable spectral densities which admit factorization in the class D^u_v × D_ε 135

Chapter 7. Interpolation Problem for Stochastic Sequences with Stationary nth Increments Observed with Non-stationary Noise139

7.1. The classical interpolation problem in the case of non-stationary noise 140

7.2. Minimax (robust) method of interpolation 148

7.3. Least favorable spectral densities in the class D⁻_0,μ× D⁻_0,μ 150

7.4. Least favorable spectral densities in the class D⁻_M,μ×D⁻_M,μ 153

Chapter 8. Filtering Problem for Stochastic Sequences with Stationary nth Increments Observed with Non-stationary Noise155

8.1. The classical filtering problem in the case of non-stationary noise 156

8.2. Minimax filtering based on observations with non-stationary noise 170

8.3. Least favorable spectral densities in the class D⁰_f × D⁰_g 174

8.4. Least favorable spectral densities which admit factorizations in theclass D⁰_f × D⁰_g 175

8.5. Least favorable spectral densities in the class D^u_v × D_ε 177

8.6. Least favorable spectral densities which admit factorizations in the class D^u_v × D_ε 178

Chapter 9. Stationary Increments of Continuous Time Stochastic Processes: Spectral Representation181

Chapter 10. Extrapolation Problem for Stochastic Processes with Stationary nth Increments187

10.1. Hilbert space projection method of extrapolation 187

10.2. Minimax (robust) method extrapolation 205

10.3. Least favorable spectral densities in the class D⁰_f × D⁰_g 208

10.4. Least favorable spectral density in the class D⁰_f 210

10.5. Least favorable spectral density which admits factorization in the class D⁰_f 211

10.6. Least favorable spectral densities in the class D^u_v × D_ε 213

10.7. Least favorable spectral densities which allow factorization in the class D_δ 215

Chapter 11. Interpolation Problem for Stochastic Processes with Stationary nth Increments217

11.1. Hilbert space projection method of interpolation 217

11.2. Minimax (robust) method of interpolation 226

11.3. Least favorable spectral densities in the class D⁰_f × D⁰_g 229

11.4. Least favorable spectral density in the class D⁰_f 230

11.5. Least favorable spectral densities in the class D⁰_1/f × D⁰_1/g 231

11.6. Least favorable spectral density in the class D⁰_1/f 233

11.7. Least favorable spectral densities in the class D^u_v × D_ε 234

11.8. Least favorable spectral density in the class D^u_v 235

11.9. Least favorable spectral density in the class D_2ε 236

Chapter 12. Filtering Problem for Stochastic Processes with Stationary nth Increments239

12.1. Hilbert space projection method of filtering 239

12.2. Minimax (robust) method of filtering 246

12.3. Least favorable spectral densities in the class D⁰_f × D⁰_g 248

12.4. Least favorable spectral densities in the class D^u_v × D_ε 250

Problems to Solve 253

Appendix 259

References 267

Index 281

Maksym Luz is Deputy Local Chief Actuary and Risk Officer at BNP Paribas Cardif, Ukraine.

Mikhail Moklyachuk is Full Professor at the Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine.