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The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

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  • JULIE LYNG FORMAN
  • MICHAEL SØRENSEN

Abstract

The Pearson diffusions form a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. Explicit optimal martingale estimating functions are found. The discussion covers GMM, quasi-likelihood, non-linear weighted least squares estimation and likelihood inference too. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions and Pearson stochastic volatility models. For the non-Markov models, explicit optimal prediction-based estimating functions are found. The estimators are shown to be consistent and asymptotically normal. Copyright (c) Board of the Foundation of the Scandinavian Journal of Statistics 2008.

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Article provided by Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association in its journal Scandinavian Journal of Statistics.

Volume (Year): 35 (2008)
Issue (Month): 3 ()
Pages: 438-465

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Handle: RePEc:bla:scjsta:v:35:y:2008:i:3:p:438-465

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Cited by:
  1. Aleksandar Mijatovic & Paul Schneider, 2009. "Empirical asset pricing with nonlinear risk premia," Papers 0911.0928, arXiv.org.
  2. Yuichi Nagahara, 2008. "A Method of Calculating the Downside Risk by Multivariate Nonnormal Distributions," Asia-Pacific Financial Markets, Springer, Springer, vol. 15(3), pages 175-184, December.
  3. Yuichi Nagahara, 2011. "Using Nonnormal Distributions to Analyze the Relationship Between Stock Returns in Japan and the US," Asia-Pacific Financial Markets, Springer, Springer, vol. 18(4), pages 429-443, November.
  4. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, Springer, vol. 16(4), pages 711-740, October.
  5. Asger Lunde & Anne Floor Brix, 2013. "Estimating Stochastic Volatility Models using Prediction-based Estimating Functions," CREATES Research Papers 2013-23, School of Economics and Management, University of Aarhus.
  6. Almut E. D. Veraart & Luitgard A. M. Veraart, 2009. "Stochastic volatility and stochastic leverage," CREATES Research Papers 2009-20, School of Economics and Management, University of Aarhus.
  7. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, Elsevier, vol. 176(2), pages 93-111.

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