Parameter estimation in nonlinear stochastic differential equations

Abstract We discuss the problem of parameter estimation in nonlinear stochastic differential equations (SDEs) based on sampled time series. A central message from the theory of integrating SDEs is that there exist in general two time scales, i.e. that of integrating these equations and that of sampling. We argue that therefore, maximum likelihood estimation is computationally extremely expensive. We discuss the relation between maximum likelihood and quasi maximum likelihood estimation. In a simulation study, we compare the quasi maximum likelihood method with an approach for parameter estimation in nonlinear SDEs that disregards the existence of the two time scales.

[1]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[2]  Theo Gasser,et al.  Smoothing Techniques for Curve Estimation , 1979 .

[3]  Egerton Smith Eyres.  XXXIV. On the simplification of mathematical analyses:— a paper read to the Liverpool Philosophical Society , 1814 .

[4]  Balth van der Pol Jun. Doct.Sc..  LXXXV. On oscillation hysteresis in a triode generator with two degrees of freedom , 1922 .

[5]  D. Florens-zmirou.  Approximate discrete-time schemes for statistics of diffusion processes , 1989 .

[6]  L. J. Comrie,et al.  Recent Progress in Scientific Computing , 1944 .

[7]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[8]  William H. Press,et al.  Numerical recipes , 1990 .

[9]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[10]  Hermann Haken,et al.  Unbiased determination of forces causing observed processes , 1992 .

[11]  Jens Timmer.  Modeling Noisy Time Series: Physiological Tremor , 1998 .

[12]  Stefan Letott,et al.  Statistical Inference Based on the Likelihood , 1999, Technometrics.

[13]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[14]  Baake,et al.  Fitting ordinary differential equations to chaotic data. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[15]  D. Florens-zmirou,et al.  Estimation of the coefficients of a diffusion from discrete observations , 1986 .

[16]  C. Heyde,et al.  Quasi-likelihood and its application , 1997 .

[17]  Peter E. Kloeden,et al.  The Numerical Solution of Nonlinear Stochastic Dynamical Systems: a Brief Introduction , 1991 .

[18]  H. Leung,et al.  STOCHASTIC TRANSIENT OF A NOISY VAN DER POL OSCILLATOR , 1995 .

[19]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[20]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[21]  Klaus Schulten,et al.  Effect of noise and perturbations on limit cycle systems , 1991 .

[22]  Hermann Haken,et al.  Unbiased estimate of forces from measured correlation functions, including the case of strong multiplicative noise , 1992 .

[23]  R. Fox,et al.  On the amplification of molecular fluctuations for nonstationary systems: hydrodynamic fluctuations for the Lorenz model , 1993 .

[24]  M. Sørensen,et al.  Martingale estimation functions for discretely observed diffusion processes , 1995 .

[25]  Gouesbet.  Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[26]  R. Friedrich,et al.  Analysis of data sets of stochastic systems , 1998 .

[27]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[28]  Johannes P. Schlöder,et al.  Modelling the fast fluorescence rise of photosynthesis , 1992 .

[29]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[30]  Tohru Ozaki,et al.  Comparative study of estimation methods for continuous time stochastic processes , 1997 .