Advertisement

Statistics and Computing

, Volume 22, Issue 1, pp 205–217 | Cite as

Approximation of transition densities of stochastic differential equations by saddlepoint methods applied to small-time Ito–Taylor sample-path expansions

  • S. P. PrestonEmail author
  • Andrew T. A. Wood
Article

Abstract

Likelihood-based inference for parameters of stochastic differential equation (SDE) models is challenging because for most SDEs the transition density is unknown. We propose a method for estimating the transition density that involves expanding the sample path as an Ito–Taylor series, calculating the moment generating function of the retained terms in the Ito–Taylor expansion, then employing a saddlepoint approximation. We perform a numerical comparison with two other methods similarly based on small-time expansions and discuss the pros and cons of our new method relative to other approaches.

Keywords

Ito–Taylor expansion Saddlepoint approximation Stochastic differential equation Transition density 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aït-Sahalia, Y.: Transition densities for interest rate and other nonlinear diffusions. J. Finance 54, 1361–1395 (1999) CrossRefGoogle Scholar
  2. Aït-Sahalia, Y.: Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70, 223–262 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Aït-Sahalia, Y.: Closed-form likelihood expansions for multivariate diffusions. Ann. Stat. 36, 906–937 (2008) zbMATHCrossRefGoogle Scholar
  4. Aït-Sahalia, Y., Yu, J.: Saddlepoint approximations for continuous-time Markov processes. J. Econom. 134, 507–551 (2006) CrossRefGoogle Scholar
  5. Beskos, A., Papaspiliopoulos, O., Roberts, G.O., Fearnhead, P.: Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). J. R. Stat. Soc. B 68, 333–382 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  6. Butler, R.W.: Saddlepoint Approximations with Applications. Cambridge University Press, Cambridge (2007) zbMATHCrossRefGoogle Scholar
  7. Daniels, H.E.: Saddlepoint approximations in statistics. Ann. Math. Stat. 25, 631–650 (1954) MathSciNetzbMATHCrossRefGoogle Scholar
  8. Dacunha-Castelle, D., Florens-Zmirou, D.: Estimation of the coefficient of a diffusion from discrete observations. Stochastics 19, 263–284 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  9. Durham, G.B., Gallant, A.R.: Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econ. Stat. 20, 297–338 (2002) MathSciNetCrossRefGoogle Scholar
  10. Florens-Zmirou, D.: Approximate discrete time schemes for statistics of diffusion processes. Statistics 21, 547–557 (1989) MathSciNetCrossRefGoogle Scholar
  11. Kessler, M.: Estimation of an ergodic diffusion from discrete observations. Scand. J. Stat. 24, 211–229 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  12. Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992) zbMATHGoogle Scholar
  13. Lindström, E.: Estimating parameters in diffusion processes using an approximate maximum likelihood approach. Ann. Oper. Res. 151, 269–288 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  14. Pedersen, A.R.: A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat. 22, 55–71 (1995) zbMATHGoogle Scholar
  15. Pennisi, L.L.: Elements of Complex Variables, 2nd edn. Holt, Reinhart and Winstoin, New York (1976) zbMATHGoogle Scholar
  16. Prakasa-Rao, B.L.S.: Asymptotic theory for non-linear least squares estimator for diffusion process. Math. Operationforsch. Stat. Ser. Stat. 14, 195–2009 (1983) MathSciNetzbMATHGoogle Scholar
  17. Prakasa-Rao, B.L.S.: Statistical inference from sampled data for stochastic processes. Contemp. Math. 80, 249–284 (1988) MathSciNetGoogle Scholar
  18. Shepp, L.A.: On the integral of the absolute value of the pinned Wiener process. Ann. Probab. 10, 234–239 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  19. Shoji, I., Ozaki, T.: Estimation for nonlinear stochastic differential equations by a local linearization method. Stoch. Anal. Appl. 16, 733–752 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Stramer, O., Yan, J.: On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation. J. Comput. Graph. Stat. 16, 672–691 (2007) MathSciNetCrossRefGoogle Scholar
  21. Yoshida, N.: Estimation for diffusion processes from discrete observations. J. Multivar. Anal. 41, 220–242 (1992) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Division of Statistics, School of Mathematical SciencesUniversity of NottinghamNottinghamUK

Personalised recommendations