Least-squares estimators based on the Adams method for stochastic differential equations with small Lévy noise
Tóm tắt
We consider stochastic differential equations (SDEs) driven by small Lévy noise with some unknown parameters and propose a new type of least-squares estimators based on discrete samples from the SDEs. To approximate the increments of a process from the SDEs, we shall use not the usual Euler method but the Adams method, that is, a well-known numerical approximation of the solution to the ordinary differential equation appearing in the limit of the SDE. We show the consistency of the proposed estimators and the asymptotic distribution in a suitable observation scheme. We also show that our estimators can be better than the usual LSE based on the Euler method in the finite sample performance.
Tài liệu tham khảo
Applebaum, D. (2009). Lévy processes and stochastic calculus, second ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge.
Butcher, J. C. (2016). Numerical methods for ordinary differential equations (3rd ed.). Chichester: John Wiley & Sons Ltd.
Davis, P. J. (1963). Interpolation and approximation. republication, with minor corrections, of the original, with a new preface and bibliography (p. 1975). New York: Dover Publications Inc.
Evans, L. C. (2010). Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI.
Genon-Catalot, V. (1990). Maximum contrast estimation for diffusion processes from discrete observations. Statistics, 21(1), 99–116.
Hairer, E. Nørsett, S. P. and Wanner, G. (1993). Solving ordinary differential equations. i. Nonstiff problems, second ed., Springer Series in Computational Mathematics, vol. 8, Springer.
Hairer, E. & Wanner, G. Solving ordinary differential equations. ii. stiff and differential-algebraic problems, second revised ed., Springer Series in Computational Mathematics, vol. 14, Springer.
Iserles, A. (2008). A first course in the numerical analysis of differential equations, 2 ed., Cambridge Texts in Applied Mathematics, Cambridge University Press.
Laredo, C. F. (1990). A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process. The Annals of Statistics, 18(3), 1158–1171.
Long, H., Shimizu, Y., & Sun, W. (2013). Least squares estimators for discretely observed stochastic processes driven by small Lévy noises. Journal of Multivariate Analysis, 116, 422–439.
Long, H., Ma, C., & Shimizu, Y. (2017). Least squares estimators for stochastic differential equations driven by small lévy noises. Stochastic Process Application, 127(5), 1475–1495.
Mao, X. (2008). Stochastic differential equations and applications (2nd ed.). Chichester: Horwood Publishing Limited.
Ogihara, T., & Yoshida, N. (2011). Quasi-likelihood analysis for the stochastic differential equation with jumps. Statistical Inference for Stochastic Processes, 14(3), 189–229.
Shimizu, Y. (2017). Threshold estimation for stochastic processes with small noise. Scandinavian Journal of Statistics, 44(4), 951–988.
Sørensen, M., & Uchida, M. (2003). Small-diffusion asymptotics for discretely sampled stochastic differential equations. Bernoulli, 9(6), 1051–1069.
Uchida, M. (2004). Estimation for discretely observed small diffusions based on approximate martingale estimating functions. Scandinavian Journal of Statistics, 31(4), 553–566.
van der Vaart, A. W. (1998). Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics (Vol. 3). Cambridge: Cambridge University Press.