ARTICLE
ABSTRACT
Estimation methods for stochastic differentia equations driver by discretely sampled continuous diffusion processes may be split into two categories: maximum likelihood methods and methods based on the general method of moments. Usually, one does not know neither likelihood function nor theoretical moments of diffusion process and cannot construct estimators. Therefore many methods was developed to approximating unknown transition density. The aim of article is to compare properties of selected approaches, indicate their merits and limitations.
KEYWORDS
stochastic differential equations, diffusion processes, maximum likelihood estimation
REFERENCES
Ait-Sahalia Y., (1999), Transition Densities for Interest Rate and Other Nonlinear Diffusions, The Journal of Finance, 54 (4), 1361–1395.
Ait-Sahalia Y., (2002), Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach, Econometrica, 70 (1), 223–262.
Ait-Sahalia Y., (2008), Closed-form Likelihood Expansions for Multivariate Diffusions, The Annals of Statistics, 36 (2), 906–937.
Billingsley P., (1961), Statistical Inference for Markov Processes (Vol. 2), Chicago: University of Chicago Press.
Broze L., Scaillet O., Zakoian J. M., (1998), Quasi-Indirect Inference for Diffusion Processes, Econometric Theory, 14 (02), 161–186.
Chacko G., Viceira L. M., (2003), Spectral GMM Estimation of Continuous-Time Processes, Journal of Econometrics, 116 (1), 259–292.
Chan K. C., Karolyi G. A., Longstaff F. A., Sanders A. B., (1992), An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, The Journal of Finance, 47 (3), 1209–1227.
Christensen B. J., Poulsen R., Sorensen M., (2001), Optimal Inference for Diffusion Processes with Applications to the Short Rate of Interest, (No. 102), working paper.
DiPietro M., (2001), Bayesian Inference for Discretely Sampled Diffusion Processes with Financial Applications, praca doktorska, Department of Statistics, Carnegie-Mellon University.
Durham G. B., Gallant A. R., (2002), Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion processes, Journal of Business & Economic Statistics, 20 (3), 297–338.
Elerian O., (1998), A Note on the Existence of a Closed Form Conditional Transition Density for the Milstein Scheme, Economics Discussion Paper, W18.
Elerian O., Chib S., Shephard N., (2001), Likelihood Inference for Discretely Observed Nonlinear Diffusions, Econometrica, 959–993.
Florens-Zmirou D., (1989), Approximate Discrete-Time Schemes for Statistics of Diffusion Processes, Statistics: A Journal of Theoretical and Applied Statistics, 20 (4), 547–557.
Gallant A. R., Tauchen G., (1996), Which Moments to Match?, Econometric Theory, 12 (04), 657–681.
Gourieroux C., Monfort A., Renault E., (1993), Indirect Inference, Journal of Applied Econometrics, 8, S85–S85.
Hansen L. P., (1982), Large Sample Properties of Generalized Method of Moments Estimators, Econometrica: Journal of the Econometric Society, 1029–1054.
Hurn A. S., Lindsay K. A., (1999), Estimating the Parameters of Stochastic Differential Equations, Mathematics and Computers in Simulation, 48 (4), 373–384.
Jensen B., Poulsen R., (2002), Transition Densities of Diffusion Processes: Numerical Comparison of Approximation Techniques, The Journal of Derivatives, 9 (4), 18–32.
Kolmogorov A. N., (1931), On Analytical Methods in the Theory of Probability. Math. Ann, 104, 415–458.
Kostrzewski M., (2004), Bayesowska estymacja parametrów dyskretnie obserwowalnych procesów dyfuzji (na przykładzie modelu CIR), Przegląd Statystyczny, 3, 129–139.
Kostrzewski M., (2006), Bayesowska analiza finansowych szeregów czasowych modelowanych procesami dyfuzji, AGH Uczelniane Wydawnictwa Naukowo-Dydaktyczne.
Lo A. W., (1988), Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data, Econometric Theory, 4 (02), 231–247.
Ozaki T., (1992), A Bridge Between Nonlinear Time Series Models and Nonlinear Stochastic Dynamical Systems: a Local Linearization Approach, Statistica Sinica, 2 (1), 113–135.
Pedersen A. R., (1995a), A New Approach to Maximum Likelihood Estimation for Stochastic Differential Equations Based on Discrete Observations, Scandinavian Journal of Statistics, 55–71.
Pedersen A. R., (1995b), Consistency and Asymptotic Normality of an Approximate Maximum Likelihood Estimator for Discretely Observed Diffusion Processes. Bernoulli, 257–279.
Shoji I., Ozaki T., (1998), Estimation for Nonlinear Stochastic Differential Equations by a Local Linearization Method, Stochastic Analysis and Applications, 16 (4), 733–752.
