无迭代积分的Milstein格式的总变分界

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":"无迭代积分的Milstein格式的总变分界","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":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":null,\"pages\":null},\"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}","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

摘要

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。
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
Total variation bound for Milstein scheme without iterated integrals
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.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 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
Investigating the ecological fallacy through sampling distributions constructed from finite populations Joint application of the Monte Carlo method and computational probabilistic analysis in problems of numerical modeling with data uncertainties Choice of a constant in the expression for the error of the Monte Carlo method Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling A gradient method for high-dimensional BSDEs
×
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