Optimal Iterative Threshold Kernel Estimation of Jump-Diffusion Processes
In this dissertation, we study the non-parametric estimation of a jump-diffusion process through an iterative threshold kernel method. Jump-diffusion processes are widely used in Mathematical Finance to model asset prices. With the availability of high frequency data, more attention has been paid to the estimation of the spot volatility and the detection of jumps, especially under non-parametric settings. It turns out that the two problems are closely related. We first deal with the spot volatility estimation of continuous stochastic processes via kernel methods. Under some mild conditions, which are verified by most processes used in Finance, we characterize the asymptotic behavior of the Mean Squared Error (MSE) of the kernel estimator. Using the MSE as an objective function, a novel plug-in type bandwidth selection method is proposed and the optimal kernel function is obtained under various settings. We also establish the Central Limit Theorems of the proposed kernel estimators. Next, we consider threshold methods for the jump detection of jump-diffusion processes. Using the number of jump misclassifications as an objective function, we extend a second order approximation of the optimal threshold, first reported in Nisen 2013, to processes with time-dependent coefficients and nonzero drift. The formula depends on the spot volatility, jump density at the origin, and spot jump intensity. We proceed to propose a threshold-kernel based estimator to estimate the jump density at the origin, and show that the optimal threshold for this end is still proportional to the modulus of continuity of the Brownian motion. Finally, we conclude with an iterative threshold kernel estimation method to jointly estimate the spot volatility and detect jumps. Extensive simulation studies show superior performance of the proposed approach.
Zhang, Purdue University.
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