{"title":"Total variation bound for Milstein scheme without iterated integrals","authors":"Toshihiro Yamada","doi":"10.2139/ssrn.4333285","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 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.","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/10.2139/ssrn.4333285","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.