On the monitoring error of the supremum of a normal jump diffusion process

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Abstract

We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the form a 0/N 1/2+ a1/N 3/2+⋯b 1/N+b 2/N 2+b 4/N 4+⋯, where N is the number of monitoring intervals. We obtain explicit expressions for the coefficients {a 0, a 1,..., b 1, b 2,...}. In particular, a0 is proportional to the value of the Riemann zeta function at 1/2, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

Original languageEnglish (US)
Pages (from-to)1021-1034
Number of pages14
JournalJournal of Applied Probability
Volume48
Issue number4
DOIs
StatePublished - Dec 2011

Keywords

  • Discrete monitoring
  • Euler-Maclaurin formula
  • Lerch transcendent
  • Normal jump diffusion process
  • Riemann zeta function
  • Spitzer's identity
  • Supremum

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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