Paper Review: “Parameter Estimation in a

Stochastic Drift Hidden Markov Model with a Cap”

byJ. 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)