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42 results

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  • ON THE EXPECTED UNIFORM ERROR OF BROWNIAN MOTION APPROXIMATED BY THE LÉVY–CIESIELSKI CONSTRUCTION

  • Part of
  • BRUCE BROWN, MICHAEL GRIEBEL, FRANCES Y. KUO, IAN H. SLOAN
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 109 / Issue 3 / June 2024
    Published online by Cambridge University Press:
    24 August 2023, pp. 581-593
    Print publication:
    June 2024
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  • The Brownian bridge or Lévy–Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. We focus on the uniform error. In particular, we show constructively that at level N, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order $\mathcal {O}(\sqrt {\ln d/d})$, matching the known orders. We apply the results to an option pricing example.

  • The heritability of Nematodirus battus fecal egg counts

  • Saeid Nikbin, Fazel Almasi, Dalal Alenizi, Caitlin Jenvey, Sarah Sloan, Sarah Preston, David Piedrafita, Nicholas Jonsson, Michael Stear
  • Journal:
    Parasitology / Volume 149 / Issue 4 / April 2022
    Published online by Cambridge University Press:
    26 January 2022, pp. 555-561
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  • Although Nematodirus battus is a serious threat to the health and survival of young lambs, there are few options to control this parasite. Bayesian Monte Carlo Markov Chain modelling with a zero-inflated Poisson distribution was used to estimate the heritability of egg counts in both June and July for each of five consecutive cohorts of 200 Scottish Blackface lambs. In one of the 10 analyses, the results failed the diagnostic tests. In seven of the analyses, there was no convincing evidence that the variation in egg counts was heritable. In the 2 years of high infection, the heritability was approximately 0.4 in June but the estimates lacked precision and the 95% highest posterior density credible intervals ranged from just above zero to 0.7. Selective breeding for resistance to N. battus will be difficult because genetically resistant or susceptible lambs cannot be consistently identified by phenotypic markers.

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