Advanced

Estimating parameters in diffusion processes using an approximate maximum likelihood approach

Lindström, Erik LU (2007) In Annals of Operations Research 151(1). p.269-288
Abstract
We present an approximate Maximum Likelihood estimator for univariate Ito stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker-Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences. The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from... (More)
We present an approximate Maximum Likelihood estimator for univariate Ito stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker-Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences. The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from the Cox-Ingersoll-Ross model and a non-linear extension of the Chan-Karolyi-Longstaff-Sanders model. The estimates are similar to the Maximum Likelihood estimates when these can be calculated and converge to the true Maximum Likelihood estimates as the accuracy of the numerical scheme is increased. The estimator is also compared to two benchmarks; a simulation-based estimator and a Crank-Nicholson scheme applied to the Fokker-Planck equation, and the proposed estimator is still competitive. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Cox-Ingersoll-Ross model, non-linear CKLS model, Crank-Nicholson scheme, approximate likelihood function, Durham-Gallant estimator
in
Annals of Operations Research
volume
151
issue
1
pages
269 - 288
publisher
Kluwer
external identifiers
  • wos:000244450800013
  • scopus:33847265348
ISSN
1572-9338
DOI
10.1007/s10479-006-0126-4
language
English
LU publication?
yes
id
25da36e0-70dc-4c7f-9fb8-8b182a2fd566 (old id 674359)
date added to LUP
2007-12-13 12:54:52
date last changed
2017-10-01 04:42:45
@article{25da36e0-70dc-4c7f-9fb8-8b182a2fd566,
  abstract     = {We present an approximate Maximum Likelihood estimator for univariate Ito stochastic differential equations driven by Brownian motion, based on numerical calculation of the likelihood function. The transition probability density of a stochastic differential equation is given by the Kolmogorov forward equation, known as the Fokker-Planck equation. This partial differential equation can only be solved analytically for a limited number of models, which is the reason for applying numerical methods based on higher order finite differences. The approximate likelihood converges to the true likelihood, both theoretically and in our simulations, implying that the estimator has many nice properties. The estimator is evaluated on simulated data from the Cox-Ingersoll-Ross model and a non-linear extension of the Chan-Karolyi-Longstaff-Sanders model. The estimates are similar to the Maximum Likelihood estimates when these can be calculated and converge to the true Maximum Likelihood estimates as the accuracy of the numerical scheme is increased. The estimator is also compared to two benchmarks; a simulation-based estimator and a Crank-Nicholson scheme applied to the Fokker-Planck equation, and the proposed estimator is still competitive.},
  author       = {Lindström, Erik},
  issn         = {1572-9338},
  keyword      = {Cox-Ingersoll-Ross model,non-linear CKLS model,Crank-Nicholson scheme,approximate likelihood function,Durham-Gallant estimator},
  language     = {eng},
  number       = {1},
  pages        = {269--288},
  publisher    = {Kluwer},
  series       = {Annals of Operations Research},
  title        = {Estimating parameters in diffusion processes using an approximate maximum likelihood approach},
  url          = {http://dx.doi.org/10.1007/s10479-006-0126-4},
  volume       = {151},
  year         = {2007},
}