Monitoring Error of the Supremum of a Normal Jump Diffusion Process

14 Pages Posted: 13 Jul 2011

See all articles by Ao Chen

Ao Chen

University of Illinois at Urbana-Champaign

Liming Feng

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering

Renming Song

affiliation not provided to SSRN

Date Written: June 26, 2011

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 i.i.d. normal jump sizes. The monitoring error is of the form $a_0/N^{1/2}$ $ a_1/N^{3/2}$ $ \cdots$ $ b_1/N$ $ b_2/N^2$ $ b_4/N^4$ $ \cdots$, where $N$ is the number of monitoring intervals. We obtain explicit expressions for the coefficients $\{a_0,a_1,\cdots,b_1,b_2,\cdots\}$. In particular, $a_0$ 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.

Keywords: normal jump diffusion process, supremum, discrete monitoring, Spitzer's identity, Euler-Maclaurin formula, Riemann zeta function, Lerch transcendent

Suggested Citation

Chen, Ao and Feng, Liming and Song, Renming, Monitoring Error of the Supremum of a Normal Jump Diffusion Process (June 26, 2011). Journal of Applied Probability, Forthcoming, Available at SSRN: https://ssrn.com/abstract=1883768

Ao Chen

University of Illinois at Urbana-Champaign ( email )

601 E John St
Champaign, IL Champaign 61820
United States

Liming Feng (Contact Author)

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering ( email )

104 S. Mathews Avenue
Urbana, IL 61801
United States

Renming Song

affiliation not provided to SSRN ( email )

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