Electronic Journal of Probability

Predictable projections of conformal stochastic integrals: an application to Hermite series and to Widder's representation

Matteo Casserini and Freddy Delbaen

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In this article, we study predictable projections of stochastic integrals with respect to the conformal Brownian motion, extending the connection between powers of the conformal Brownian motion and the corresponding Hermite polynomials. As a consequence of this result, we then investigate the relation between analytic functions and $L^p$-convergent series of Hermite polynomials. Finally, our results are applied to Widder's representation for a class of Brownian martingales, retrieving a characterization for the moments of Widder's measure.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 22, 14 pp.

Accepted: 14 March 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60G46: Martingales and classical analysis 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Predictable projections stochastic integrals conformal Brownian motion Hermite polynomials Brownian martingales Widder's representation

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Casserini, Matteo; Delbaen, Freddy. Predictable projections of conformal stochastic integrals: an application to Hermite series and to Widder's representation. Electron. J. Probab. 17 (2012), paper no. 22, 14 pp. doi:10.1214/EJP.v17-1883. https://projecteuclid.org/euclid.ejp/1465062344

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  • Alili, Larbi; Patie, Pierre. Boundary-crossing identities for diffusions having the time-inversion property. J. Theoret. Probab. 23 (2010), no. 1, 65–84.
  • Davis, Burgess. Brownian motion and analytic functions. Ann. Probab. 7 (1979), no. 6, 913–932.
  • Dellacherie, Claude; Meyer, Paul-André. Probabilities and potential. B. Theory of martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. xvii+463 pp. ISBN: 0-444-86526-8.
  • Getoor, Ronald K.; Sharpe, Michael J. Conformal martingales. Invent. Math. 16 (1972), 271–308.
  • Hille, Einar. A class of reciprocal functions. Ann. of Math. (2) 27 (1926), no. 4, 427–464.
  • Hille, Einar. Contributions to the theory of Hermitian series. Duke Math. J. 5, (1939). 875–936.
  • Hille, Einar. Contributions to the theory of Hermitian series. II. The representation problem. Trans. Amer. Math. Soc. 47, (1940). 80–94.
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8.
  • Ledoux, Michel. Isoperimetry and Gaussian analysis. Lectures on probability theory and statistics (Saint-Flour, 1994), 165–294, Lecture Notes in Math., 1648, Springer, Berlin, 1996.
  • Mansuy, Roger; Yor, Marc. Aspects of Brownian motion. Universitext. Springer-Verlag, Berlin, 2008. xiv+195 pp. ISBN: 978-3-540-22347-4.
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5.
  • Pollard, Harry. The mean convergence of orthogonal series. II. Trans. Amer. Math. Soc. 63, (1948). 355–367.
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Version 2.1. Corrected third printing. Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. ISBN: 3-540-00313-4.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7.
  • Widder, David V. Positive temperatures on an infinite rod. Trans. Amer. Math. Soc. 55, (1944). 85–95.