Asymptotic statistical Inference theory for stochastic process

1.对满足混合条件的连续时间马氏过程给出了一个渐近展开式。用泛函的Malliavin协方差的非退化条件代替离散时间环境下的条件型Cramer条件。这种方法适用于随机微分方程。有效性问题同时得到解决.将结果1应用于统计参数模型，并推导出M估计量的展开式。对于扩散模型，本文对公式中的系数进行了完整的描述。这与下面的结果1和结果5一起，已经成为该领域的基础文献，并且今天它被应用于各种统计问题。对于小噪声扩散过程的统计模型，我们导出了模型选择的信息准则.当证券价格由一般的非线性随机微分方程描述时，计算一个随机微分方程的价格是一个困难的问题， ...更多信息 派生词然而，可以通过渐近展开技术来近似它。这种方法是作者提出的，最近许多作者都在追求这种方法。结果4处理了一种实际情况，即安全性方程中存在未知参数。本文评估了估计量替代对期权价格逼近的影响，并提出了一种正确的逼近方法.对于具有长记忆随机系数的随机微分方程，通常的混合条件被打破，因此需要一种新的方法来推导渐近展开式。结果5引入了部分混合的概念，并成功地导出了渐近展开式（当然是有效的）。利用支撑定理给出了一种实用的简便方法来证明扩展泛函的Malliavin协方差的局部非退化性。这个新装置使我们能够非常容易地导出展开式，就像i.i.d.模型。条件期望是极限定理中最不规则的泛函。我们应用Malliavin演算来推导条件期望的渐近展开式。将这一结果应用于带跳随机微分方程和滤波问题。少

1. We presented an asymptotic expansion to a continuous-time Markov process satisfying the mixing condition. The conditional type Cramer condition in discrete-time setting was replaced by the nondegeneracy condition of the Malliavin covariance of the functional. This method applied to stochastic differential equations. The validity problem was at the same time solved.2. Result 1 was applied to statistical parametric models, and expansions for M-estimators were derived. For diffusion models, this paper completely described the coefficients in the formulae. This, together with Result 1 and Result 5 below, has been the fundamental literature in this field, and it is applied to various statistical problems today.3. For statistical models of diffusion processes with small noises, we derived so-called information criteria for model selection.4. When the security price is described by a general nonlinear stochastic differential equation, it is a difficult problem to compute the value of a der … More ivative. However, it is possible to approximate it by means of the asymptotic expansion technique. This method was introduced by the author and recently many authors pursuit this method. Result 4 treated a practical situation, that is, the equation of the security has unknown parameters. This paper assessed the effects of substitution of estimators to the approximation of option prices, and proposed a correct way to the good approximation.5. Since for a stochastic differential equation with random coefficients of long memory, the usual mixing condition was brokenn, a new methodology is necessary to derive asymptotic expansions. Result 5 introduced the notion of partial mixing and successfully derived asymptotic expansions (of course with validity). Moreover, it showed a practical convenient method with the support theorem to verify the local nondegeneracy of the Malliavin covariance of the expanded functional. This new device enables us to derive expansions very easily, like i.i.d. models.6. Conditional expectation is a most irregular functional in limit theorems. We applied the Malliavin calculus to derive asymptotic expansion of the conditional expectation. This result was applied to the stochastic differential equation with jumps and filtering problems. Less