Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

doi:10.1007/s11464-017-0663-y

Front. Math. China

2017, Vol. 12

Issue (6) : 1441-1455 https://doi.org/10.1007/s11464-017-0663-y

RESEARCH ARTICLE

Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

Peiyan LI¹, Wei GU²(

)

¹. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
². School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

Download: PDF(235 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

A method based on higher-order partial differential equation (PDE) numerical scheme are proposed to obtain the transition cumulative distribution function (CDF) of the diffusion process (numerical differentiation of the transition CDF follows the transition probability density function (PDF)), where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numerical tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.

Keywords Kolmogorov partial differential equations transition probability density function transition cumulative distribution function compact difference scheme

Corresponding Author(s): Wei GU

Issue Date: 27 November 2017

Cite this article:

Peiyan LI,Wei GU. Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application[J]. Front. Math. China, 2017, 12(6): 1441-1455.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0663-y
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I6/1441

1	Ait-SahaliaY. Maximum likelihood estimation of discretely sampled diffusions: a closed form approximation approach.Econometrica, 2002, 70(1): 223–262 https://doi.org/10.1111/1468-0262.00274
2	Ait-SahaliaY. Closed-form likelihood expansions for multivariate diffusions.Ann Statist, 2008, 36(2): 906–937 https://doi.org/10.1214/009053607000000622
3	BrandtM, Santa-ClaraP. Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets.J Financ Econ, 2002, 63(2): 161–210 https://doi.org/10.1016/S0304-405X(01)00093-9
4	ChanK C, KarolyiG A, LongstaffF A, SandersA B. An empirical comparison of alternative models of the short-term interest rate.J Finance, 1992, 47(3): 1209–1227 https://doi.org/10.1111/j.1540-6261.1992.tb04011.x
5	CoxJ C, IngersollJ E, RossS A. A theory of the term structure of interest rates.Econometrica, 1985, 53(2): 385–407 https://doi.org/10.2307/1911242
6	DurhamG B, GallantA R. Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes.J Bus Econom Statist, 2002, 20(3): 297–316 https://doi.org/10.1198/073500102288618397
7	HurnA S, JeismanJ, LindsayK A. Transition densities of diffusion processes: a new approach to solving the Fokker-planck equation.J Deriv, 2007, 14(4): 86–94 https://doi.org/10.3905/jod.2007.686424
8	JensenB, PoulsenR. Transition densities of diffusion processes: numerical comparison of approximation techniques.J Deriv, 2002, 9(4): 18–32 https://doi.org/10.3905/jod.2002.319183
9	KaratzasI, ShreveS. Brownian Motion and Stochastic Calculus.New York: Springer, 1992
10	LapidusL, PinderG F. Numerical Solution of Partial Differential Equations in Science and Engineering.New York: John Wiley, 1999 https://doi.org/10.1002/9781118032961
11	LoA W. Maximum likelihood estimation of generalized Itô processes with discretely sampled data.Econom Theory, 1988, 4(2): 231–247 https://doi.org/10.1017/S0266466600012044
12	NielsenJ N, MadsenH, YoungP C. Parameter estimation in stochastic differential equations: An overview.Annu Rev Control, 2000, 24: 83–94 https://doi.org/10.1016/S1367-5788(00)90017-8
13	SorensenH. Parametric inference for diffusion processes observed at discrete points in time: a survey.Int Stat Rev, 2004, 72(3): 337–354 https://doi.org/10.1111/j.1751-5823.2004.tb00241.x
14	PedersenA R. A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations.Scand J Stat, 1995, 22(1): 55–71
15	SunZ Z. The Numerical Methods for Partial Equations.Beijing: Science Press, 2005 (in Chinese)

Viewed

Full text

Abstract

Cited

Shared

Discussed