Identification of pseudo ito processes from its SFCs
Identification of pseudo ito processes from its SFCs
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从 SFC 中识别伪 ito 进程
DOI：
10.1016/j.bulsci.2013.12.003
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发表时间：
2014
影响因子：
1.3
通讯作者：
S.Ogawa & H.Uemura
中科院分区：
文献类型：
--
作者：
R. Cada;S. Chiba;K. Ozeki;P. Vrana;K. Yoshimoto;Y. Sawada and A. Tanikawa;S.Ogawa & H.Uemura
设X t是由布朗运动W.驱动的Skorokhod型非因果Itô过程，也就是说，形式为d X t= B（t，ω）d t+ a（t，ω）d W t的随机过程，其中a（t）d W t被理解为Skorokhod积分。对于这样的Itô过程X t，我们考虑由Fn（d X）=<$0 1 en（t）<$d X t，en（t）= exp对d X t进行微分的傅里叶系数Fn（d X）。（2 π− 1 n t）（n∈ Z），我们关心的是基本问题：能否从随机傅立叶系数{Fn（dx），n∈ Z}的完备集合中识别出两个参数a（ω，ω），B（ω，ω）。在本文中，我们在非因果演算的框架中研究这个问题，就像我们在以前的文章中所做的那样（Ogawa，2013; Ogawa and Uemura，in press），我们给出了肯定的答案，并给出了参数a（ω，ω），B（t，ω）的具体重建方案。我们的结果将为P. Malliavin等人提出的波动率估计的傅里叶级数方法的理论背景提供另一种解释。（Malliavin and Mancino，2002; Malliavin and Thalmaier，2009）。
Let X t be a noncausal Itô process of Skorokhod type driven by the Brownian motion W., that is, a stochastic process of the form d X t= b (t, ω) d t+ a (t, ω) d W t where the term a (⋅) d W t is understood as Skorokhod integral. For such an Itô process X t we consider the Fourier coefficient F n (d X) of the differential d X t by F n (d X)=∫ 0 1 e n (t)¯ d X t, e n (t)= exp (2 π− 1 n t)(n∈ Z) and we are concerned with the elementary question: whether we can identify the two parameters a (⋅, ω), b (⋅, ω) from the complete set of the stochastic Fourier coefficients {F n (d X), n∈ Z}. In this note we study this problem in a framework of noncausal calculus, as we did in the previous articles (Ogawa, 2013; Ogawa and Uemura, in press), and we give an affirmative answer with a concrete scheme for the reconstruction of the parameters a (⋅, ω), b (t, ω). Our result will give another light to the theoretical background of the method of Fourier series for the volatility estimation proposed by P. Malliavin et al.(Malliavin and Mancino, 2002; Malliavin and Thalmaier, 2009).