First exit densities of Brownian motion through one-sided moving boundaries

  • Christel Jennen
  • Hans Rudolf Lerche


Let {ψa; a ε ℝ} be a sequence of curved boundaries which tend to infinity as a increases. Let
$$T_a = \inf \{ t > 0|W(t) \geqq \psi _a (t)\} $$
where W(t) denotes the standard Brownian motion. Under regularity conditions on the boundaries uniform approximations for the first exit densities of Ta are derived. The consequences for upper and lower class functions are discussed. The approximations for the first exit densities of Brownian motion with drift, which are also derived, lead to uniform approximations for the power functions of sequential tests. The quality of the approximations is demonstrated by some figures.


Stochastic Process Brownian Motion Probability Theory Power Function Mathematical Biology 
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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Christel Jennen
    • 1
  • Hans Rudolf Lerche
    • 1
  1. 1.Institut für Angewandte Mathematik, Sonderforschungsbereich 123Universität HeidelbergHeidelbergFederal Republic of Germany

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