Paper Review: “Parameter Estimation in a Stochastic Drift Hidden Markov Model with a Cap” by J. Hernandez, D. Saunders & L. Seco Anatoliy Swishchuk Math & Comp Finance Lab, Dept of Math & Stat, U of C “Lunch at the Lab” Talk February 3, 2006
Model
Interpretation of the Model and Specification
Difference Between this Model and Pilipovich Model
Mixing Coefficients
Mixing Lemma
Transition Probabilities and Space
Mixing Coefficients Through P_t
Infinitesimal Generator
Spectral Gap Inequality
Spectral Gap
Definition of Hidden Markov Model
Ergodicity and Mixing
Stationarity and Hidden Markov Model
Hidden Markov Model
Assumptions I-III
Assumption IV
Main Result
Follows from the Birkhoff’s Ergodic Result
An Example: the Ornstein- Uhlenbeck Model
Transformation
Matrix Form
Another Expression
Gaussian Distribution
Transition Probability
Limits for Mean and for Covariance Matrix
Gaussian Stationary Distribution
Convergence
To Study the Law of the Process Y
Process y(t+h)
Joint Distribution of Y_t and Y_{t+h}
Estimation of Parameters
Final Calculation of Parameters
References
References (cntd)