Frontiers of Mathematics in China >
Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application
Received date: 01 Nov 2016
Accepted date: 18 Sep 2017
Published date: 27 Nov 2017
Copyright
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.
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]. Frontiers of Mathematics in China, 2017 , 12(6) : 1441 -1455 . DOI: 10.1007/s11464-017-0663-y
1 |
Ait-SahaliaY. Maximum likelihood estimation of discretely sampled diffusions: a closed form approximation approach.Econometrica, 2002, 70(1): 223–262
|
2 |
Ait-SahaliaY. Closed-form likelihood expansions for multivariate diffusions.Ann Statist, 2008, 36(2): 906–937
|
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
|
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
|
5 |
CoxJ C, IngersollJ E, RossS A. A theory of the term structure of interest rates.Econometrica, 1985, 53(2): 385–407
|
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
|
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
|
8 |
JensenB, PoulsenR. Transition densities of diffusion processes: numerical comparison of approximation techniques.J Deriv, 2002, 9(4): 18–32
|
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
|
11 |
LoA W. Maximum likelihood estimation of generalized Itô processes with discretely sampled data.Econom Theory, 1988, 4(2): 231–247
|
12 |
NielsenJ N, MadsenH, YoungP C. Parameter estimation in stochastic differential equations: An overview.Annu Rev Control, 2000, 24: 83–94
|
13 |
SorensenH. Parametric inference for diffusion processes observed at discrete points in time: a survey.Int Stat Rev, 2004, 72(3): 337–354
|
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)
|
/
〈 | 〉 |