  In this paper, we propose a new threshold-kernel jump-detection method for
jump-diffusion processes, which iteratively applies thresholding and kernel
methods in an approximately optimal way to achieve improved finite-sample
performance. We use the expected number of jump misclassifications as the
objective function to optimally select the threshold parameter of the jump
detection scheme. We prove that the objective function is quasi-convex and
obtain a new second-order infill approximation of the optimal threshold in
closed form. The approximate optimal threshold depends not only on the spot
volatility, but also the jump intensity and the value of the jump density at
the origin. Estimation methods for these quantities are then developed, where
the spot volatility is estimated by a kernel estimator with thresholding and
the value of the jump density at the origin is estimated by a density kernel
estimator applied to those increments deemed to contain jumps by the chosen
thresholding criterion. Due to the interdependency between the model parameters
and the approximate optimal estimators built to estimate them, a type of
iterative fixed-point algorithm is developed to implement them. Simulation
studies for a prototypical stochastic volatility model show that it is not only
feasible to implement the higher-order local optimal threshold scheme but also
that this is superior to those based only on the first order approximation
and/or on average values of the parameters over the estimation time period.
