On the monitoring error of the supremum of a normal jump diffusion process

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.