Total variation bound for Milstein scheme without iterated integrals

IF 0.8 Q3 STATISTICS & PROBABILITY Monte Carlo Methods and Applications Pub Date : 2023-05-26 DOI:10.1515/mcma-2023-2007
Toshihiro Yamada
{"title":"Total variation bound for Milstein scheme without iterated integrals","authors":"Toshihiro Yamada","doi":"10.1515/mcma-2023-2007","DOIUrl":null,"url":null,"abstract":"Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msubsup> <m:mi>X</m:mi> <m:mi>T</m:mi> <m:mi>ε</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>T</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>Mil</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>/</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> d_{\\mathrm{TV}}(X_{T}^{\\varepsilon},\\bar{X}_{T}^{\\varepsilon,\\mathrm{Mil},(n)})\\leq C\\varepsilon^{2}/n and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msubsup> <m:mi>X</m:mi> <m:mi>T</m:mi> <m:mi>ε</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>T</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>EM</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mi>ε</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> d_{\\mathrm{TV}}(X_{T}^{\\varepsilon},\\bar{X}_{T}^{\\varepsilon,\\mathrm{EM},(n)})\\leq C\\varepsilon/n , where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> </m:math> d_{\\mathrm{TV}} is the total variation distance, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>X</m:mi> <m:mi>ε</m:mi> </m:msup> </m:math> X^{\\varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>Mil</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> \\bar{X}^{\\varepsilon,\\mathrm{Mil},(n)} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>EM</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> \\bar{X}^{\\varepsilon,\\mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"2 1","pages":"0"},"PeriodicalIF":0.8000,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monte Carlo Methods and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/mcma-2023-2007","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ( X T ε , X ¯ T ε , Mil , ( n ) ) C ε 2 / n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/n and d TV ( X T ε , X ¯ T ε , EM , ( n ) ) C ε / n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n , where d TV d_{\mathrm{TV}} is the total variation distance, X ε X^{\varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) \bar{X}^{\varepsilon,\mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) \bar{X}^{\varepsilon,\mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.
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
无迭代积分的Milstein格式的总变分界
摘要本文给出了随机微分方程米尔斯坦格式的新结果。我们证明(i) Milstein格式在全变分意义上是弱逼近,并且在非交换向量场下由布朗运动的二阶多项式给出,而不使用迭代积分;(ii)在渐近意义上,Milstein格式的精度优于Euler-Maruyama格式。特别是,我们证明了d TV减去(X T ε, X¯T ε, Mil,(n))≤C减去ε 2/n d_ {\mathrm{TV}} ({X_T}^ {\varepsilon}, \bar{X} _T{^ }{\varepsilon, \mathrm{Mil},(n)})\leq C \varepsilon ^{2}/n和d TV减去(X T ε, X¯T ε, EM,(n))≤C减去ε /n d_ {\mathrm{TV}} ({X_T}^ {\varepsilon}, \bar{X} _T{^ }{\varepsilon, \mathrm{EM},(n)})\leq C \varepsilon /n,其中,d TV d_ {\mathrm{TV}}为总变差距离,X ε X^ {\varepsilon}为小参数方程的随机微分方程的解,X¯ε, Mil,(n) \bar{X} ^ {\varepsilon, \mathrm{Mil},(n)}和X¯ε, EM,(n)\bar{X} ^ {\varepsilon, \mathrm{EM},(n)}分别为无迭代积分的Milstein格式和Euler-Maruyama格式。在计算方面,该方案可以通过简单的模拟来估计概率分布函数,而无需计算lsamvy面积。数值算例验证了该方法的有效性。
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
Monte Carlo Methods and Applications
Monte Carlo Methods and Applications STATISTICS & PROBABILITY-
CiteScore
1.20
自引率
22.20%
发文量
31
期刊最新文献
Asymmetric kernel method in the study of strong stability of the PH/M/1 queuing system Random walk on spheres method for solving anisotropic transient diffusion problems and flux calculations Strong approximation of a two-factor stochastic volatility model under local Lipschitz condition On the estimation of periodic signals in the diffusion process using a high-frequency scheme Stochastic simulation of electron transport in a strong electrical field in low-dimensional heterostructures
×
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