This paper is on the issue of finding a closed-form likelihood approximation of diffusion processes and rearranging the Hermite expansion in the order of the power of the observational time interval. We propose an algorithm that calculates the coefficients of the rearranged expansion that Aït-Sahalia (2002) suggested. That is, a general expression of the coefficients is provided explicitly, which as far as we know has not been given in the existing literature. We also introduce a reduced form of the rearranged expansion and call it as the delta expansion in the paper. Moreover, we are able to obtain an explicit expansion of the moments in the order of the power of the observational time interval. We examine the delta expansion and the Hermite expansion without rearrangement numerically to find that the delta expansion has such advantageous features as the order of the error bound can be more effectively attained. It is also found that our expansion gives a comparable numerical accuracy of the approximation to the expansion Aït-Sahalia (1999) suggested, while making any symbolic computation unnecessary.
Bibliographical noteFunding Information:
This work was supported by the Sogang University Research Grant of 2012 (Y.D. Lee), by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 , E.-K. Lee), and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0026070 , S. Song).
- Diffusion model
- Edgeworth expansion
- Hermite expansion
- Likelihood estimation
- Transition density