{"title":"由驱动的模型的统计推断𝑛-阶分数布朗运动","authors":"Hicham Chaouch, H. Maroufy, Mohamed Omari","doi":"10.1090/tpms/1185","DOIUrl":null,"url":null,"abstract":"<p>We consider the following stochastic integral equation <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X left-parenthesis t right-parenthesis equals mu t plus sigma integral Subscript 0 Superscript t Baseline phi left-parenthesis s right-parenthesis d upper B Subscript upper H Superscript n Baseline left-parenthesis s right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>X</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:mi>t</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:msubsup>\n <mml:mo>∫<!-- ∫ --></mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mi>t</mml:mi>\n </mml:msubsup>\n <mml:mi>φ<!-- φ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>s</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:msubsup>\n <mml:mi>B</mml:mi>\n <mml:mi>H</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>s</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">X(t)=\\mu t + \\sigma \\int _0^t \\varphi (s) dB_H^n(s)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>t</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">t\\geq 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"phi\">\n <mml:semantics>\n <mml:mi>φ<!-- φ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\varphi</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a known function and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Subscript upper H Superscript n\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mi>B</mml:mi>\n <mml:mi>H</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">B^n_H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\n <mml:semantics>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\mu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma squared\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\sigma ^2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, then we formulate explicitly a least squares estimator for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\n <mml:semantics>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\mu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and an estimator for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma squared\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\sigma ^2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Statistical inference for models driven by 𝑛-th order fractional Brownian motion\",\"authors\":\"Hicham Chaouch, H. Maroufy, Mohamed Omari\",\"doi\":\"10.1090/tpms/1185\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the following stochastic integral equation <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X left-parenthesis t right-parenthesis equals mu t plus sigma integral Subscript 0 Superscript t Baseline phi left-parenthesis s right-parenthesis d upper B Subscript upper H Superscript n Baseline left-parenthesis s right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>X</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mi>μ<!-- μ --></mml:mi>\\n <mml:mi>t</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mi>σ<!-- σ --></mml:mi>\\n <mml:msubsup>\\n <mml:mo>∫<!-- ∫ --></mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mi>t</mml:mi>\\n </mml:msubsup>\\n <mml:mi>φ<!-- φ --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>s</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mi>d</mml:mi>\\n <mml:msubsup>\\n <mml:mi>B</mml:mi>\\n <mml:mi>H</mml:mi>\\n <mml:mi>n</mml:mi>\\n </mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>s</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">X(t)=\\\\mu t + \\\\sigma \\\\int _0^t \\\\varphi (s) dB_H^n(s)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"t greater-than-or-equal-to 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>t</mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">t\\\\geq 0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"phi\\\">\\n <mml:semantics>\\n <mml:mi>φ<!-- φ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\varphi</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is a known function and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper B Subscript upper H Superscript n\\\">\\n <mml:semantics>\\n <mml:msubsup>\\n <mml:mi>B</mml:mi>\\n <mml:mi>H</mml:mi>\\n <mml:mi>n</mml:mi>\\n </mml:msubsup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">B^n_H</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\">\\n <mml:semantics>\\n <mml:mi>n</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"mu\\\">\\n <mml:semantics>\\n <mml:mi>μ<!-- μ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mu</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"sigma squared\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>σ<!-- σ --></mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\sigma ^2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, then we formulate explicitly a least squares estimator for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"mu\\\">\\n <mml:semantics>\\n <mml:mi>μ<!-- μ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mu</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and an estimator for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"sigma squared\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>σ<!-- σ --></mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\sigma ^2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.</p>\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-05-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tpms/1185\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1185","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Statistical inference for models driven by 𝑛-th order fractional Brownian motion
We consider the following stochastic integral equation X(t)=μt+σ∫0tφ(s)dBHn(s)X(t)=\mu t + \sigma \int _0^t \varphi (s) dB_H^n(s), t≥0t\geq 0, where φ\varphi is a known function and BHnB^n_H is the nn-th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both μ\mu and σ2\sigma ^2, then we formulate explicitly a least squares estimator for μ\mu and an estimator for σ2\sigma ^2 by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.