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 form a₀ / N^1/2 + a₁ / N^3/2 + · · · + b₁ / N + b₂ / N² + b₄ / N⁴ + · · ·, where N is the number of monitoring intervals. We obtain explicit expressions for the coefficients {a₀, a₁, . . . , b₁, b₂, . . .}. In particular, a₀ 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.