Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences

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 Representation 1 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 D0f 24 2.4. Least favorable spectral densities which admit factorization in the class D0f 25 2.5. Least favorable spectral density in the class Duv 29 2.6. Least favorable spectral density which admits factorization in the class Duv 29 Chapter 3. Interpolation Problem for Stochastic Sequences with Stationary nth Increments 31 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 Noise 53 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 D0f x D0g 80 4.5. Least favorable spectral densities which admit factorization in the class D0f x D0g 82 4.6. Least favorable spectral densities in the class Duv x D 84 4.7. Least favorable spectral densities which admit factorization in the class Duv x D 86 Chapter 5. Interpolation Problem for Stochastic Sequences with Stationary nth Increments Based on Observations with Stationary Noise 89 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,fx D 0,g 100 5.5. Least favorable spectral densities in the class D2 1x D1 2 103 Chapter 6. Filtering Problem of Stochastic Sequences with Stationary nth Increments Based on Observations with Stationary Noise 107 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 D0f x D0g 129 6.5. Least favorable spectral densities which admit factorization in the class D0f x D0g 131 6.6. Least favorable spectral densities in the class Duv x D 134 6.7. Least favorable spectral densities which admit factorization in the class Duv x D 135 Chapter 7. Interpolation Problem for Stochastic Sequences with Stationary nth Increments Observed with Non-stationary Noise 139 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, x D 0, 150 7.4. Least favorable spectral densities in the class D M, xD M, 153 Chapter 8. Filtering Problem for Stochastic Sequences with Stationary nth Increments Observed with Non-stationary Noise 155 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 D0f x D0g 174 8.4. Least favorable spectral densities which admit factorizations in theclass D0f x D0g 175 8.5. Least favorable spectral densities in the class Duv x D 177 8.6. Least favorable spectral densities which admit factorizations in the class Duv x D 178 Chapter 9. Stationary Increments of Continuous Time Stochastic Processes: Spectral Representation 181 Chapter 10. Extrapolation Problem for Stochastic Processes with Stationary nth Increments 187 10.1. Hilbert space projection method of extrapolation 187 10.2. Minimax (robust) method extrapolation 205 10.3. Least favorable spectral densitie