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Re ected Brownian Bridge area conditioned on its local

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15 pages
Re ected Brownian Bridge area conditioned on its local time at the origin Philippe Chassaing Guy Louchard y March 28, 2002 Abstract Using some properties of the Airy functions, we analyze the re ected Brownian Bridge area W b conditioned on its local time b at the origin. We give a closed form expression of the Laplace trans- form of W b , a recurrence equation for the moments, leading to an eÆcient computation algorithm and an asymptotic form for the density f(x; b) of W b for x _ 0. 1 Introduction Throughout this paper, the standard Brownian motion (BM) will be denoted by x(t). Other classical BM are the re ected BM: x + (t) := jx(t)j, the Brownian Bridge (BB) on [0; 1]: B(t), the re ected BB on [0; 1]: B + (t), the Brownian Excursion: e(t). The local time of x(t) at a, will be denoted by t + (t; a). Following Janson [9, remark 2.3], we dene W b := R 1 0 B + (t)dt (area of the re ected BB), conditioned on having a local time at the origin equal to b.

  • brownian bridge

  • airy function

  • laplace transform

  • conditioned

  • function iterates

  • bb area

  • generalized airy

  • standard brownian


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