Parameter Estimation for Multiscale Diffusions
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We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized single-scale model to such multiscale data. We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified. We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly.
The ideas are studied in the context of thermally activated motion in a two-scale potential. However the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data.
Keywordsparameter estimation multiscale diffusions stochastic differential equations homogenization maximum likelihood subsampling
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- 3.O. E. Barndorff-Nielsen, P. R. Hansen, A. Lunde and N. Shephard, Designing realised kernels to measure the ex-post variation of equity in the presence of noise. Preprint (2006).Google Scholar
- 4.I. V. Basawa and B. L. S. Prakasa Rao, Statistical inference for stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers] (London, 1980).Google Scholar
- 5.A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic analysis of periodic structures. (North-Holland, Amsterdam, 1978).Google Scholar
- 6.C. P. Calderon, Fitting effective diffusion models to data associated with a glassy potential: Estimation, classical inference procedures and some heuristics. SIAM Multiscale Modeling and Simulation,to appear.Google Scholar
- 7.F. Campillo and A. Piatnitski, Effective diffusion in vanishing viscosity. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), volume 31 of Stud. Math. Appl. (pp. 133–145, North-Holland, Amsterdam, 2002).Google Scholar
- 11.J.-P. Fouque, G. Papanicolaou R. Sircar, and K. Solna, Short time scale in S and P volatility. J. Comp. Finance 6(4):1–23 (2003).Google Scholar
- 13.M. Freidlin, Functional integration and partial differential equations, volume 109 of Annals of Mathematics Studies. (Princeton University Press, Princeton, NJ, 1985).Google Scholar
- 18.I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. (Springer-Verlag, New York, second edition 1991).Google Scholar
- 19.P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics (New York). (Springer-Verlag, Berlin, 1992).Google Scholar
- 20.R. S. Liptser and A. N. Shiryaev, Statistics of random processes. I, volume 5 of Applications of Mathematics (New York). (Springer-Verlag, Berlin, 2001).Google Scholar
- 21.X. Mao, Stochastic differential equations and their applications. (Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester, 1997).Google Scholar
- 23.S. Olla, Homogenization of diffusion processes in random fields. (Lecture Notes, 1994).Google Scholar
- 25.G. A. Pavliotis and A. M. Stuart, An introduction to Multiscale Methods: Averaging and Homogenization (Lecture Notes, 2006).Google Scholar
- 26.D. Revuz and M. Yor, Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. (Springer-Verlag, Berlin, third edition, 1999).Google Scholar