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2 results

  • Sojourns and extremes of a diffusion process on a fixed interval

  • Simeon M. Berman
  • Journal:
    Advances in Applied Probability / Volume 14 / Issue 4 / December 1982
    Published online by Cambridge University Press:
    01 July 2016, pp. 811-832
    Print publication:
    December 1982

  • Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

  • On the maximum of a stationary independent increment process

  • Sheldon M. Ross
  • Journal:
    Journal of Applied Probability / Volume 9 / Issue 3 / June 1972
    Published online by Cambridge University Press:
    14 July 2016, pp. 677-680
    Print publication:
    June 1972

  • A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.