×

zbMATH — the first resource for mathematics

Spectral methods for identifying scalar diffusions. (English) Zbl 0962.62094
Summary: This paper shows how to identify nonparametrically scalar stationary diffusions from discrete-time data. The local evolution of the diffusion is characterized by a drift and diffusion coefficient along with the specification of boundary behavior. We recover this local evolution from two objects that can be inferred directly from discrete-time data: the stationary density and a conveniently chosen eigenvalue-eigenfunction pair of the conditional expectation operator over a unit interval of time. This construction also lends itself to a spectral characterization of the over-identifying restrictions implied by a scalar diffusion model of a discrete-time Markov process.

MSC:
62M99 Inference from stochastic processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
62G99 Nonparametric inference
62P05 Applications of statistics to actuarial sciences and financial mathematics
47N30 Applications of operator theory in probability theory and statistics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Ait-Sahalia, Y.: Nonparametric pricing of interest rate derivative securities. Econometrica 64, No. 3, 527-560 (1996) · Zbl 0844.62094
[2] Ait-Sahalia, Y., 1996b. Do interest rates really follow continuous-time markov diffusions. Manuscript.
[3] Banon, G.: Nonparametric identification for diffusion processes. SIAM journal of control and optimization 16, 380-395 (1978) · Zbl 0404.93045
[4] Birkhoff, G.; Rota, G. C.: Ordinary differential equations, 4th edn. (1989) · Zbl 0183.35601
[5] Bochner, S.: Harmonic analysis and the theory of probability. (1960) · Zbl 0106.05203
[6] Bouc, R.; Pardoux, E.: Asymptotic analysis of p.d.e.s with wide-band noise disturbances, and expansion of the moments. Stochastic analysis and applications 2, 369-422 (1984) · Zbl 0574.60066
[7] Clark, P. K.: A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, No. 1, 135-155 (1973) · Zbl 0308.90011
[8] Conley, T., Hansen, L.P., Luttmer, E.G.J., Scheinkman, J., 1997. Short-term interest rates as subordinated diffusions. The Review of Financial Studies, forthcoming.
[9] Demoura, S.G., 1993. Theory and application of transition operator to the analysis of economic time series. Necessary and sufficient conditions for nonlinearities in economic dynamics. Aliasing problem. Manuscript.
[10] Duffie, D., Kan, R., 1993. A yield-factor model of interest rates. Manuscript. · Zbl 1065.91507
[11] Duffie, D., Glynn, P., 1996. Estimation of continuous-time markov processes sampled at random time intervals. Manuscript. · Zbl 1142.62390
[12] Ethier, S. N.; Kurtz, T. G.: Markov processes – characterization and convergence. (1986) · Zbl 0592.60049
[13] Florens, J.-P., Renault, E., Touzi, N., 1995. Testing for embeddability by stationary scalar diffusions. Manuscript.
[14] Frydman, H., Singer, B., 1979. Total positivity and the embedding problem for markov chains. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 86, pp. 339–344. · Zbl 0411.60071
[15] Hansen, L. P.; Scheinkman, J. A.: Back to the future: generating moment implications for continuous-time Markov processes. Econometrica 63, No. 4, 767-804 (1995) · Zbl 0834.60083
[16] Hartman, P., 1973. Ordinary Differential Equations, Baltimore. · Zbl 0281.34001
[17] Karatzas, I.; Shreve, S.: Browian motion and stochastic calculus. (1988) · Zbl 0638.60065
[18] Karlin, S.: Total positivity. (1968) · Zbl 0219.47030
[19] Karlin, S.; Mcgregor, J. L.: A characterization of birth and death processes. Proceedings of the Academy of science 45, 375-379 (1959) · Zbl 0204.20701
[20] Karlin, S.; Taylor, H. M.: A second course in stochastic processes. (1981) · Zbl 0469.60001
[21] Kessler, M., Sorensen, M., 1996. Estimating equations based on eigenfunctions for a discretely observed diffusion process. Unpublished.
[22] Mandl, P.: Analytical treatment of one-dimensional Markov processes. (1968) · Zbl 0179.47802
[23] Pazy, A., 1983. Semigroups of linear operators and applications to partial differential equations (New York: Springer-Verlag). · Zbl 0516.47023
[24] Revuz, Yor, M., 1994. Coontinuous Martingales and Browian Motion. Springer, Berlin. · Zbl 0804.60001
[25] Singer, B.; Spillerman, S.: Fitting stochastic models to longitudinal survey data – some examples in the social sciences. Bulletin of the international statistical institute 67, 283-300 (1977)
[26] Weidmann J., 1980. Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics, Vol. 68. Springer, Berlin. · Zbl 0434.47001
[27] Weidmann J., 1987. Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, vol. 1258. Springer, Berlin. · Zbl 0647.47052
[28] Wong E., 1964. The construction of a class of stationary markoff processes. In: Belleman, R., (Ed.), Sixteenth Symposium in Applied Mathematics – Stochastic Processes in Mathematical Physics and Engineering, American Mathematical Society, Providence, RI, pp. 264–276. · Zbl 0139.34406
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.