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On Probability Characteristics of "Downfalls" in a Standard Brownian Motion

Abstract : For a Brownian motion $B=(B_t)_{t\le 1}$ with $B_0=0$, {\bf E}$B_t=0$, {\bf E}$B_t^2=t$ problems of probability distributions and their characteristics are considered for the variables $$ \begin{array}{c} {\mathbb D} =\displaystyle\sup_{0\le t\le t'\le 1}(B_t-B_{t'}),\qquad {\mathbb D}_1=B_\sigma-\inf_{\sigma\le t'\le 1}B_{t'}, \\ {\mathbb D}_2=\displaystyle\sup_{0\le t\le\sigma'}B_{t}-B_{\sigma'}, \end{array} $$ where $\sigma$ and $\sigma'$ are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on $[0,1]$ (i.e., $B_\sigma=\sup_{0\le t\le 1}B_t$, $B_{\sigma'}=\inf_{0\le t'\le 1}B_{t'}$).
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Raphaël Douady, A.N. Shiryaev, Marc yor. On Probability Characteristics of "Downfalls" in a Standard Brownian Motion. Theory of Probability and Its Applications c/c of Teoriia Veroiatnostei i Ee Primenenie, Society for Industrial and Applied Mathematics, 2000, 44 (1), pp.29-38. ⟨10.1137/S0040585X97977306⟩. ⟨hal-01477104⟩



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