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On the Monitoring Error of the Supremum of a Normal Jump Diffusion Process

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations Mathematical finance Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ao Chen ,
Liming Feng  and
Renming Song
Ao Chen*
Affiliation:
University of Illinois at Urbana-Champaign
Liming Feng*
Affiliation:
University of Illinois at Urbana-Champaign
Renming Song*
Affiliation:
University of Illinois at Urbana-Champaign
*
∗ Postal address: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.
∗∗∗ Postal address: Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Email address: fenglm@illinois.edu
∗ Postal address: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.
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Abstract

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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 + a 1 / 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, a 0 is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

Keywords

Normal jump diffusion processsupremumdiscrete monitoringSpitzer's identityEuler-Maclaurin formulaRiemann zeta functionLerch transcendent

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60J75: Jump processes 91G60: Numerical methods (including Monte Carlo methods) 65C20: Models, numerical methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1021 - 1034
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Research partially supported by the National Science Foundation, under grants CMMI-0927367 and CMMI-1029846.

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