Processes of Wright-Fisher Type

  • Aurélien Alfonsi
Part of the Bocconi & Springer Series book series (BS, volume 6)


The main focus of this book is on affine diffusions and their simulation. In this chapter, we go slightly beyond this scope in the sense that the processes that we consider are not affine. They however have a clear connection with affine processes and belong to the class of Polynomial processes introduced by Cuchiero et al. [34] that also includes Affine diffusions. We first present Wright-Fisher processes that are well known in biology to model the frequency of a gene. These processes are directly related to Cox-Ingersoll-Ross diffusions, and we explain how it is possible to get second order schemes for these processes by reusing the second order schemes that we have developed for the CIR diffusion.


Correlation Matrice Jacobi Polynomial Order Scheme Infinitesimal Generator Martingale Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC (1964)Google Scholar
  2. 3.
    Ahdida, A., Alfonsi, A.: A mean-reverting SDE on correlation matrices. Stoch. Process. Appl. 123(4), 1472–1520 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 27.
    Chen, L., Stroock, D.W.: The fundamental solution to the Wright-Fisher equation. SIAM J. Math. Anal. 42(2), 539–567 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 34.
    Cuchiero, C., Keller-Ressel, M., Teichmann, J.: Polynomial processes and their applications to mathematical finance. Financ. Stoch. 16, 711–740 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 50.
    Epstein, C.L., Mazzeo, R.: Wright-Fisher diffusion in one dimension. SIAM J. Math. Anal. 42(2), 568–608 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 51.
    Etheridge, A.: Some Mathematical Models from Population Genetics. Lecture Notes in Mathematics, vol. 2012. Springer, Heidelberg (2011). Lectures from the 39th Probability Summer School held in Saint-Flour (2009)Google Scholar
  7. 68.
    Gourieroux, C., Jasiak, J.: Multivariate Jacobi process with application to smooth transitions. J. Econ. 131(1–2), 475–505 (2006)CrossRefMathSciNetGoogle Scholar
  8. 83.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)Google Scholar
  9. 84.
    Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic [Harcourt Brace Jovanovich Publishers], New York (1981)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Aurélien Alfonsi
    • 1
  1. 1.CERMICSEcole Nationale des Ponts et ChausséesChamps-sur-MarneFrance

Personalised recommendations