×

Estimating the parameters of stochastic differential equations. (English) Zbl 0927.62082

Summary: Two maximum likelihood methods for estimating the parameters of stochastic differential equations (SDEs) from time-series data are proposed. The first is that of simulated maximum likelihood in which a nonparametric kernel is used to construct the transitional density of an SDE from a series of simulated trials. The second approach uses a spectral technique to solve the Kolmogorov equation satisfied by the transitional probability density. The exact likelihood function for a geometric random walk is used as a benchmark against which the performance of each method is measured. Both methods perform well with the spectral method returning results which are practically identical to those derived from the exact likelihood. The technique is illustrated by modelling interest rates in the UK gilts market using a fundamental one-factor term-structure equation for the instantaneous rate of interest.

MSC:

62M09 Non-Markovian processes: estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M05 Markov processes: estimation; hidden Markov models
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ait-Sohalia, Y.: Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527-560 (1996) · Zbl 0844.62094
[2] Brennan, M. J.; Schwartz, E. S.: A continuous-time approach to the pricing of bonds. J. banking finance 3, 133-155 (1979)
[3] J.Y. Campbell, A.W. Lo, A.C. MacKinlay, The Econometries of Financial Markets, Princeton University Press, Princeton, NJ, 1997 · Zbl 0927.62113
[4] Cox, J. C.; Ingersoll, J. E.; Ross, S. A.: A theory of the term structure of interest rates. Econometrica 53, 385-407 (1985) · Zbl 1274.91447
[5] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988 · Zbl 0658.76001
[6] Hall, P.; Sheather, S. J.; Jones, M. C.; Marron, J. S.: On optimal data-based bandwidth selection in kernel density estimation. Biometrika 78, 263-270 (1991) · Zbl 0733.62045
[7] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, San Diego, 1970 · Zbl 0203.50101
[8] P.E. Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 1991
[9] Merton, R.: On estimating the expected return on the market: an exploratory investigation. J. financial economics 8, 323-361 (1980)
[10] D.W. Scott, Multivariate Density Estimation: Theory, Practice and Visualization, Wiley, New York, 1992 · Zbl 0850.62006
[11] B.W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman & Hall, London, 1986 · Zbl 0617.62042
[12] Vasicek, O.: An equilibrium characterisation of the term structure. J. financial economics 5, 177-188 (1977) · Zbl 1372.91113
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.