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Proceedings of the American Mathematical Society

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Stochastic construction of new orthogonal polynomials


Author: T. F. Lin
Journal: Proc. Amer. Math. Soc. 73 (1979), 365-370
MSC: Primary 60H05
DOI: https://doi.org/10.1090/S0002-9939-1979-0518522-0
MathSciNet review: 518522
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Abstract: For $ u > - 1,t \geqslant 0,x \in R$, let $ {(1 + u)^{Q(t) + t}}{e^{ - ut}}$ satisfies $ dy(t) = uy(t)\;dQ(t)$ where $ Q(t) = P(t) - t$ is the centered Poisson process with parameter $ \lambda = 1$, (ii) $ {K_n}(t,x),n \geqslant 0$, is a system of complete orthogonal polynomials in $ {L^2}(R,dF)$ where $ F(x)$ is the distribution function of $ P(t)$, (iii) $ (n + 1){K_{n + 1}}(t,x) - (x - t - n){K_n}(t,x) + t{K_{n - 1}}(t,x) = 0,n \geqslant 1$, (iv) $ t{\Delta ^2}{K_n}(t,x) + (t - x)\Delta {K_n}(t,x) + n{K_n}(t,x) = 0,n \geqslant 0$.


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  • [1] Ludwig Arnold, Stochastic differential equations: theory and applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Translated from the German. MR 0443083
  • [2] R. Courant and D. Hilbert, Methods of mathematical physics. I, Interscience, New York, 1955.
  • [3] Hiroshi Kunita and Shinzo Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209–245. MR 0217856
  • [4] T. F. Lin, An identity for multiple integral with respect to Poisson process, Soochow J. Math. Natur. Sci. 3 (1977), 35–40. MR 0482997
  • [5] H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
  • [6] E. J. McShane, Stochastic calculus and stochastic models, Academic Press, New York-London, 1974. Probability and Mathematical Statistics, Vol. 25. MR 0443084

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0518522-0
Keywords: Orthogonal polynomials, stochastic integral, difference equation
Article copyright: © Copyright 1979 American Mathematical Society