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2 results

  • On ℝd-valued peacocks

  • Francis Hirsch, Bernard Roynette
  • Journal:
    ESAIM: Probability and Statistics / Volume 17 / 2013
    Published online by Cambridge University Press:
    21 May 2013, pp. 444-454
    Print publication:
    2013

  • In this paper, we consider ℝd-valued integrable processes which are increasing in the convex order, i.e. ℝd-valued peacocks in our terminology. After the presentation of some examples, we show that an ℝd-valued process is a peacock if and only if it has the same one-dimensional marginals as an ℝd-valued martingale. This extends former results, obtained notably by Strassen [Ann. Math. Stat. 36 (1965) 423–439], Doob [J. Funct. Anal. 2 (1968) 207–225] and Kellerer [Math. Ann. 198 (1972) 99–122].

  • A new proof of Kellerer’s theorem

  • Francis Hirsch, Bernard Roynette
  • Journal:
    ESAIM: Probability and Statistics / Volume 16 / 2012
    Published online by Cambridge University Press:
    26 March 2012, pp. 48-60
    Print publication:
    2012

  • In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.