Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

ON THE MARTINGALE PROBLEM AND SYMMETRIC DIFFUSION IN POPULATION GENETICS

  • Choi, Won (Department of Mathematics, University of Incheon) ;
  • Joung, Yoo-Jung (Department of Mathematics, University of Incheon)
  • Received : 2009.11.10
  • Accepted : 2010.02.17
  • Published : 2010.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In allelic model $X\;=\;(x_1,\;x_2,\;\cdots,\;x_d)$, $$M_f(t)\;=\;f(p(t))\;-\;\int_0^t\;Lf(p(t))ds$$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we define $T_tf\;=\;E_{p_0}^{p^*}\;[f((P(t))]$ for $t\;{\geq}\;0$ for using a new diffusion operator $L^*$ and we show the diffusion relations between $T_t$ and diffusion operator $L^*$.

Keywords

Acknowledgement

Supported by : University of Incheon

References

  1. Won Choi, Some symmetry preserving transformation in population genetics, J. of Appl. Math. amd info., 27 (2009), 757-762.
  2. Fukushima, M, Dirichlet forms and Markov processes, Math. Libr. Ser. No., 23, North Holland Publ. Amsterdam (1980).
  3. N.Okada, On the uniqueness problem of two dimensional diffusion processes occurring in population genetics, Z.Wahr,Verw.Geb., 56 (1981), 63-74. https://doi.org/10.1007/BF00531974
  4. S.N.Either, A class of degenerate diffusion processes occurring in population genetics, Comm. Pure Appl. Math., 29 (1976), 483-493. https://doi.org/10.1002/cpa.3160290503
  5. S.N.Either, A class of infinite dimensional diffusions occurring in population genetics, Indiana Univ. J., 30 (1981), 925-935. https://doi.org/10.1512/iumj.1981.30.30068
  6. J.H.Gillespie, Natural selection for within-generation variance in offspring number, Genetics, 76 (1974), 601-606.