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The delta expansion for the transition density of diffusion models

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  • Lee, Yoon Dong
  • Song, Seongjoo
  • Lee, Eun-Kyung

Abstract

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.

Suggested Citation

  • Lee, Yoon Dong & Song, Seongjoo & Lee, Eun-Kyung, 2014. "The delta expansion for the transition density of diffusion models," Journal of Econometrics, Elsevier, vol. 178(P3), pages 694-705.
    • Handle: RePEc:eee:econom:v:178:y:2014:i:p3:p:694-705
    DOI: 10.1016/j.jeconom.2013.10.008
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    References listed on IDEAS

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    1. Seungmoon Choi, 2011. "Closed-Form Likelihood Expansions for Multivariate Time-Inhomogeneous Diffusions," School of Economics and Public Policy Working Papers 2011-26, University of Adelaide, School of Economics and Public Policy.
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    5. Osnat Stramer & Matthew Bognar & Paul Schneider, 2010. "Bayesian Inference for Discretely Sampled Markov Processes with Closed-Form Likelihood Expansions," Journal of Financial Econometrics, Oxford University Press, vol. 8(4), pages 450-480, Fall.
    6. Ai[dieresis]t-Sahalia, Yacine & Yu, Jialin, 2006. "Saddlepoint approximations for continuous-time Markov processes," Journal of Econometrics, Elsevier, vol. 134(2), pages 507-551, October.
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    9. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    10. Yu, Jialin, 2007. "Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese Yuan," Journal of Econometrics, Elsevier, vol. 141(2), pages 1245-1280, December.
    11. Bakshi, Gurdip & Ju, Nengjiu & Ou-Yang, Hui, 2006. "Estimation of continuous-time models with an application to equity volatility dynamics," Journal of Financial Economics, Elsevier, vol. 82(1), pages 227-249, October.
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    Cited by:

    1. Yang, Nian & Chen, Nan & Wan, Xiangwei, 2019. "A new delta expansion for multivariate diffusions via the Itô-Taylor expansion," Journal of Econometrics, Elsevier, vol. 209(2), pages 256-288.
    2. Wan, Xiangwei & Yang, Nian, 2021. "Hermite expansion of transition densities and European option prices for multivariate diffusions with jumps," Journal of Economic Dynamics and Control, Elsevier, vol. 125(C).
    3. Xu, Libai & Kong, Dehan & Wang, Lidan & Gu, Hong & Kenney, Toby & Xu, Ximing, 2023. "Proportional stochastic generalized Lotka–Volterra model with an application to learning microbial community structures," Applied Mathematics and Computation, Elsevier, vol. 448(C).
    4. Salima El Kolei & Fabien Navarro, 2022. "Contrast estimation for noisy observations of diffusion processes via closed-form density expansions," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 303-336, July.
    5. Chenxu Li & Yu An & Dachuan Chen & Qi Lin & Nian Si, 2016. "Efficient computation of the likelihood expansions for diffusion models," IISE Transactions, Taylor & Francis Journals, vol. 48(12), pages 1156-1171, December.

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