{"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}
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 dTV(XTε,X¯Tε,Mil,(n))≤Cε2/n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/n and dTV(XTε,X¯Tε,EM,(n))≤Cε/n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n , where dTV d_{\mathrm{TV}} is the total variation distance, Xε 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.