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 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