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1 results

  • Boundaries with negative jumps for the Brownian motion

  • C. Park
  • Journal:
    Journal of Applied Probability / Volume 22 / Issue 4 / December 1985
    Published online by Cambridge University Press:
    14 July 2016, pp. 964-970
    Print publication:
    December 1985

  • First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.