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

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 forma0/N1/2+a1/N3/2+ · · · +b1/N+b2/N2+b4/N4+ · · ·, whereNis the number of monitoring intervals. We obtain explicit expressions for the coefficients {a0,a1, …,b1,b2, …}. In particular,a0is 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.