Optimal Kernel Estimation of Spot Volatility of Stochastic Differential Equations
نویسندگان
چکیده
Abstract: The selections of the bandwidth and kernel function of a kernel estimator are of great importance in practice. This is not different in the context of spot volatility kernel estimators. In this work, a feasible method of bandwidth and kernel selection is proposed, under some mild conditions on the volatility process, which not only cover classical Brownian motion driven dynamics but also some processes driven by long-memory fractional Brownian motions. Under the proposed unifying framework, we characterize the leading order terms of the mean squared error, which in turn enables us to determine an explicit formula for the leading term of the optimal bandwidth. Central limit theorems for the estimation error are also given. A feasible plug-in type bandwidth selection procedure is then proposed, for which, as a sub-problem, a new estimator of the volatility of volatility is developed. In addition, the optimal selection of the kernel function is also investigated. For Brownian Motion type volatilities, the optimal kernel turns out to be an exponential function. For fractional Brownian motion type volatilities, numerical results to compute the optimal kernel are devised and, for the deterministic volatility case, explicit optimal kernel functions of different orders are derived. Simulation studies further confirm the good performance of the proposed methods.
منابع مشابه
Supplement to “Optimal Kernel Estimation of Spot Volatility of Stochastic Differential Equations”
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