Le Cam–Stratonovich–Boole theory for Itô diffusions

IF 0.3 Q4 STATISTICS & PROBABILITY Random Operators and Stochastic Equations Pub Date : 2023-02-28 DOI:10.1515/rose-2023-2004
J. Bishwal
{"title":"Le Cam–Stratonovich–Boole theory for Itô diffusions","authors":"J. Bishwal","doi":"10.1515/rose-2023-2004","DOIUrl":null,"url":null,"abstract":"Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\\frac{T}{n^{6/7}}\\to 0} as T → ∞ {T\\to\\infty} and n → ∞ {n\\to\\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"31 1","pages":"153 - 176"},"PeriodicalIF":0.3000,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Operators and Stochastic Equations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/rose-2023-2004","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\frac{T}{n^{6/7}}\to 0} as T → ∞ {T\to\infty} and n → ∞ {n\to\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Le Cam–Stratonovich–Itôdiffusions的布尔理论
摘要我们将Le Cam的局部渐近正态性(LAN)理论与Stratonovich随机积分的Boole近似联系起来,通过估计遍历扩散过程的非线性漂移系数中的参数,该过程满足齐次Itô随机微分方程,该过程基于离散间隔的稠密观测。在几乎缓慢增长的实验设计(ASIED)条件下(T n 6/7),获得了基于连续似然的Itô和Boole近似的近似最大似然估计量、近似最大概率估计量和近似贝叶斯估计量的渐近正态性和局部渐近极小性(在Hajek–Le Cam意义上)→ 0{\frac{T}{n^{6/7}}\到0}作为T→ ∞ {T \ to \ infty}和n→ ∞ {n\to\infty},其中T是观测时间的长度,n是观测次数)。证明了Bernstein–von Mises型定理,证明了在相同设计条件下,适当归一化和中心的近似后验分布收敛于正态分布。作为副产品,得到了在相同设计条件下条件最小二乘估计量的渐近正态性和渐近有效性。基于Itô近似的对数似然导数是鞅,但基于Boole近似的对数可能性导数不是鞅,而是前向和后向鞅的加权平均。这些新的近似比鞅近似具有更快的收敛速度。在蒙特卡洛模拟中,该方法将比欧拉和米尔斯坦近似具有优势。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Random Operators and Stochastic Equations
Random Operators and Stochastic Equations STATISTICS & PROBABILITY-
CiteScore
0.60
自引率
25.00%
发文量
24
期刊最新文献
On a reaction diffusion problem with a moving impulse on boundary Backward doubly stochastic differential equations driven by fractional Brownian motion with stochastic integral-Lipschitz coefficients Existence results for some stochastic functional integrodifferential systems driven by Rosenblatt process On Ulam type of stability for stochastic integral equations with Volterra noise Existence and uniqueness for reflected BSDE with multivariate point process and right upper semicontinuous obstacle
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1