| Many scholars paid attention on Levy process for its features with infinitely divisible, stable and independent increments long before. They devoted to the modelling of Levy process, which was started in the infinitely divisible distributions, developed to the behavior of finite jump and deepened to infinite jumps, such as Gamma, VG, NIG, CGMY. In some extent, they can be reflected the non-normal characteristics of financial data with spike and fat tail, so the subject business and academia focus on it. And then it was widely used in queuing theory, investment risk, and financial data modeling.In the study of Levy process, density estimation is the most critical work. Basing on the study of Enrique Figueroa-Lope in 2008, we use different methods based on splines which are adaptive regularized partition constant and deregularized partition constant and piecewise natural spline to achieve the nonparametric estimation of Levy density s(x) from a collection of linear modles{Sm,m ∈M) of the subspace of L2(D,η).We simulate sample paths driven by two different processes, namely Gamma, VG process in order to assess the feasibility and universal adaptability. Meanwhile, for the simulation data we try to estimate the density using the basic expansion methods. And then we apply the LSE to estimate the parameters in the Levy density function sθ(x) with the midpoint of the interval values{xi,i=1,…,m) and the values s(xi) of the nonparametric output. In addition, the parameters are compared to the ones which obtain from MLE and Enrique’. From the comparison we can find that the methods which based on splines are better for θ than MLE. Finally, this paper proposes spline-based method with S$P500 stock index and compare the results of previous studies. That demonstrates the feasibility and advantages of the research methods.Finally, the last part of the paper puts forward a thought and direction of spline-based density estimation to improve the estimation. |
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