多分数过程的局部路径行为

IF 0.4 Q4 STATISTICS & PROBABILITY Theory of Probability and Mathematical Statistics Pub Date : 2022-05-16 DOI:10.1090/tpms/1162
A. Ayache, F. Bouly
{"title":"多分数过程的局部路径行为","authors":"A. Ayache, F. Bouly","doi":"10.1090/tpms/1162","DOIUrl":null,"url":null,"abstract":"<p>Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the <italic>classical</italic> Multifractional Brownian Motion (MBM) <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{{\\mathcal {M}}(t)\\}_{t\\in \\mathbb {R}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal {H}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the well-known Fractional Brownian Motion by a deterministic function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H left-parenthesis t right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal {H}}(t)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two <italic>non-classical</italic> Gaussian multifactional processes denoted by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace upper X left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\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:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{X(t)\\}_{t\\in \\mathbb {R}}</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=\"left-brace upper Y left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mi>Y</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{Y(t)\\}_{t\\in \\mathbb {R}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>\n\n<p>In our article, under a rather weak condition on the functional parameter <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H left-parenthesis dot right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mo>⋅<!-- ⋅ --></mml:mo>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{\\mathcal {H}}(\\cdot )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we show that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{{\\mathcal {M}}(t)\\}_{t\\in \\mathbb {R}}</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=\"left-brace upper X left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\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:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{X(t)\\}_{t\\in \\mathbb {R}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as well as <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{{\\mathcal {M}}(t)\\}_{t\\in \\mathbb {R}}</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=\"left-brace upper Y left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mi>Y</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{Y(t)\\}_{t\\in \\mathbb {R}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> only differ by a part which is locally","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On local path behavior of Surgailis multifractional processes\",\"authors\":\"A. Ayache, F. Bouly\",\"doi\":\"10.1090/tpms/1162\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the <italic>classical</italic> Multifractional Brownian Motion (MBM) <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">M</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{{\\\\mathcal {M}}(t)\\\\}_{t\\\\in \\\\mathbb {R}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper H\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">H</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">{\\\\mathcal {H}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of the well-known Fractional Brownian Motion by a deterministic function <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper H left-parenthesis t right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">H</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">{\\\\mathcal {H}}(t)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two <italic>non-classical</italic> Gaussian multifactional processes denoted by <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace upper X left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\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:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{X(t)\\\\}_{t\\\\in \\\\mathbb {R}}</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=\\\"left-brace upper Y left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mi>Y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{Y(t)\\\\}_{t\\\\in \\\\mathbb {R}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\\n\\n<p>In our article, under a rather weak condition on the functional parameter <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper H left-parenthesis dot right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">H</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mo>⋅<!-- ⋅ --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">{\\\\mathcal {H}}(\\\\cdot )</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, we show that <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">M</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{{\\\\mathcal {M}}(t)\\\\}_{t\\\\in \\\\mathbb {R}}</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=\\\"left-brace upper X left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\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:msub>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{X(t)\\\\}_{t\\\\in \\\\mathbb {R}}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> as well as <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">M</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mo fence=\\\"false\\\" 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引用次数: 1

摘要

多分数过程是具有非平稳增量的随机过程,其局部正则性和自相似性随点而变化。典型的例子是经典的多分数布朗运动(MBM) {M (t)} t∈R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} (Benassi, Jaffard, lsamvy vsamhel, Peltier和Roux),它是在90年代中期建立的,只是用一个具有一定平滑性的确定性函数H (t) {\mathcal {H}}(t)代替了著名的分数阶布朗运动中的常数Hurst参数H {\mathcal {H}}。十多年后,使用一种不同的构造方法,基本上依赖于非齐次分数阶积分和微分算子,Surgailis引入了两个非经典高斯多阶过程,表示为{X(t)} t∈R \{X(t)\}_{t\in \mathbb {R}}和{Y(t)} t∈R \{Y(t)\}_{t\in \mathbb {R}}。在函数参数H(⋅){\mathcal {H}}(\cdot)的较弱条件下,我们证明了{M (t)} t∈R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}和{X(t)} t∈R \{X(t)\}_{t\in \mathbb {R}}以及{M (t)}} t∈R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}和{Y(t)} t∈R \{Y(t)\}_{t\in \mathbb {R}}只有局部的区别
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On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t R \{Y(t)\}_{t\in \mathbb {R}} .

In our article, under a rather weak condition on the functional parameter H ( ) {\mathcal {H}}(\cdot ) , we show that { M ( t ) } t R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { X ( t ) } t R \{X(t)\}_{t\in \mathbb {R}} as well as { M ( t ) } t R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t R \{Y(t)\}_{t\in \mathbb {R}} only differ by a part which is locally

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来源期刊
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