On independent times and positions for Brownian motions

  • Bernard de Meyer

    Université Henri Poincaré, Vandoeuvre lès Nancy, France
  • Bernard Roynette

    Université Henri Poincaré, Vandoeuvre lès Nancy, France
  • Pierre Vallois

    Université Henri Poincaré, Vandoeuvre lès Nancy, France
  • Marc Yor

    Université Paris VI, France

Abstract

Let , \( \big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big) \) be a one (resp. two) dimensional Brownian motion started at 0. Let be a stopping time such that \big(resp. is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if and are independent and has all exponential moments, then is constant. \item[2)] If and are independent and have all exponential moments, then and are Gaussian. \end{itemize} We also give a number of examples of stopping times , with only some exponential moments, such that and are independent, and similarly for and . We also exhibit bounded non-constant stopping times such that and are independent and Gaussian.

Cite this article

Bernard de Meyer, Bernard Roynette, Pierre Vallois, Marc Yor, On independent times and positions for Brownian motions. Rev. Mat. Iberoam. 18 (2002), no. 3, pp. 541–586

DOI 10.4171/RMI/328