JournalsrmiVol. 18, No. 3pp. 541–586

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
On independent times and positions for Brownian motions cover
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Abstract

Let (Bt;t0)(B_t ; t \ge 0), (\mboxresp.((Xt,Yt);t0))\big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big) be a one (resp. two) dimensional Brownian motion started at 0. Let TT be a stopping time such that (BtT;t0)(B_{t \wedge T} ; t \ge 0) \big(resp. (XtT;t0);(YtT;t0))(X_{t \wedge T} ; t \ge 0) ; (Y_{t \wedge T} ; t \ge 0)\big) is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if TT and BTB_T are independent and TT has all exponential moments, then TT is constant. \item[2)] If XTX_T and YTY_T are independent and have all exponential moments, then XTX_T and YTY_T are Gaussian. \end{itemize} We also give a number of examples of stopping times TT, with only some exponential moments, such that TT and BTB_T are independent, and similarly for XTX_T and YTY_T. We also exhibit bounded non-constant stopping times TT such that XTX_T and YTY_T 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