Noneparametric Estimation For Levy Processes Based On Wavelet Basis

Levy processes are the focus of the theory of stochastic processes. In the past 20 years, according to the classic B-S option pricing model, the hypothesis that continuous change of the stock price follows geometric Brown motion has been demonstrated not significant with real data by empirical researches, such as fat-tailedness. In order to do better in showing the characteristics of data and revealing the essence of changes of price, in the field of mathematical finance, Levy processes, which abandoned the geometry Brown motion assumption, have raised concerns. In the financial modeling of continuous time processes, compared to the traditional Brown motion of continuous track, Levy processes with jump can better describe the change of market price, explain the fat-tailedness, fit the statistical characteristics of financial data, as result, price the derivatives more accurately. However, among other drawbacks, the high computational intensity and numerical issues involved in calculating the Poisson stochastic integral have prevented Levy processes from being more widely used in practice especially when dealing with "high-frequency" data.This paper introduces a nonparametric estimation method of Levy density based on wavelet basis, constructs estimators for the measure of the pure-jump Levy processes, and devises the discrete form of penalized projection estimator based on Haar wavelet basis with an approximation procedure for Poisson stochastic integrals based on discrete data.In terms of model selection, with a view towards mean square error, control function and penalty function method, the objective function which is defined by Poisson stochastic integrals is constructed by two parts:loss function and penalty function which controls complexity.In terms of simulation, two kinds of Levy processes, which play important roles in the field of mathematical finance, namely Gamma Levy processes and Variance Gamma Levy processes are simulated. Penalized projection estimator based on Haar wavelet basis is constructed, and the performance of penalized projection estimator in different situations is discussed.Finally, our methods are applied to the estimation of a classic model used in risk asset pricing:Variance Gamma model, based on Haar wavelet basis.