## 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 language | English (US) |
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Pages (from-to) | 1021-1034 |

Number of pages | 14 |

Journal | Journal of Applied Probability |

Volume | 48 |

Issue number | 4 |

DOIs | |

State | Published - 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
- General Mathematics
- Statistics, Probability and Uncertainty