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Wavelet estimation of the diffusion coefficient in time dependent diffusion models
作者姓名:Ping  CHEN~
作者单位:1,2
基金项目:国家自然科学基金;教育部高等学校博士学科点专项科研基金
摘    要:The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.

收稿时间:9 March 2006
修稿时间:9 April 2007

Wavelet estimation of the diffusion coefficient in time dependent diffusion models
Ping CHEN.Wavelet estimation of the diffusion coefficient in time dependent diffusion models[J].Science in China(Mathematics),2007,50(11):1597-1610.
Authors:Ping Chen  Jin-de Wang
Institution:1. Department of Mathematics,Nanjing University,Nanjing 210093,China:School of Science,Nanjing University of Science and Technology,Nanjing 210094,China
2. Department of Mathematics,Nanjing University,Nanjing 210093,China
Abstract:The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.
Keywords:wavelet estimation  time-dependent diffusion coefficient  linear growth condition  strong consistency
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