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.

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.

Notations ix

Introduction xi

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

Chapter 2. Extrapolation Problem for Stochastic Sequences with Stationarynth 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 classD0f 24

2.4. Least favorable spectral densities which admit factorization in the classD0f 25

2.5. Least favorable spectral density in the classDuv 29

2.6. Least favorable spectral density which admits factorization in the classDuv 29

Chapter 3. Interpolation Problem for Stochastic Sequences withStationarynth Increments31

3.1. The classical method of interpolation 31

3.2. Minimax method of interpolation 41

3.3. Least favorable spectral densities in the classD0,n 43

3.4. Least favorable spectral densities in the classDM,n 47

Chapter 4. Extrapolation Problem for Stochastic Sequences with Stationarynth 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 classD0f ×D0g 80

4.5. Least favorable spectral densities which admit factorization in the classD0f ×D0g 82

4.6. Least favorable spectral densities in the classDuv × D 84

4.7. Least favorable spectral densities which admit factorization in the classDuv × D 86

Chapter 5. Interpolation Problem for Stochastic Sequences with Stationarynth 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 classD0,f× D0,g 100

5.5. Least favorable spectral densities in the classD21× D12 103

Chapter 6. Filtering Problem of Stochastic Sequences with Stationarynth 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 classD0f ×D0g 129

6.5. Least favorable spectral densities which admit factorization in the classD0f ×D0g 131

6.6. Least favorable spectral densities in the classDuv × D 134

6.7. Least favorable spectral densities which admit factorization in the classDuv × D 135

Chapter 7. Interpolation Problem for Stochastic Sequences with Stationarynth 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 classD0,× D0, 150

7.4. Least favorable spectral densities in the classDM,×DM, 153

Chapter 8. Filtering Problem for Stochastic Sequences with Stationarynth 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 classD0f ×D0g 174

8.4. Least favorable spectral densities which admit factorizations in theclassD0f ×D0g 175

8.5. Least favorable spectral densities in the classDuv × D 177

8.6. Least favorable spectral densities which admit factorizations in the classDuv × D 178

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

Chapter 10. Extrapolation Problem for Stochastic Processes with Stationarynth 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 classD0f ×D0g 208

10.4. Least favorable spectral density in the classD0f 210

10.5. Least favorable spectral density which admits factorization in the classD0f 211

10.6. Least favorable spectral densities in the classDuv × D 213

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

Chapter 11. Interpolation Problem for Stochastic Processes with Stationarynth 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 classD0f ×D0g 229

11.4. Least favorable spectral density in the classD0f 230

11.5. Least favorable spectral densities in the classD01/f ×D01/g 231

11.6. Least favorable spectral density in the classD01/f 233

11.7. Least favorable spectral densities in the classDuv × D 234

11.8. Least favorable spectral density in the classDuv 235

11.9. Least favorable spectral density in the classD2 236

Chapter 12. Filtering Problem for Stochastic Processes with Stationarynth 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 classD0f × D0g 248

12.4. Least favorable spectral densities in the classDuv × D 250

Problems to Solve 253

Appendix 259

References 267

Index 281