For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

IF 0.4 Q4 STATISTICS & PROBABILITY Theory of Probability and Mathematical Statistics Pub Date : 2021-12-07 DOI:10.1090/tpms/1157

F. Kühn, R. Schilling

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Schilling","doi":"10.1090/tpms/1157","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X equals left-parenthesis upper X Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>X</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">X=(X_t)_{t\\geq 0}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a one-dimensional Lévy process such that each <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X Subscript t\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">X_t</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript b Superscript 1\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mi>C</mml:mi>\n <mml:mi>b</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">C^1_b</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon double-struck upper R right-arrow double-struck upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</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:mo stretchy=\"false\">→</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:annotation encoding=\"application/x-tex\">f\\colon \\mathbb {R}\\to \\mathbb {R}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and exponentially bounded functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g colon double-struck upper R right-arrow left-parenthesis 0 comma normal infinity right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>g</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:mo stretchy=\"false\">→</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi mathvariant=\"normal\">∞</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">g\\colon \\mathbb {R}\\to (0,\\infty )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, such that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis upper X Subscript t Baseline right-parenthesis minus double-struck upper E f left-parenthesis upper X Subscript t Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">E</mml:mi>\n </mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f(X_t)-\\mathbb {E} f(X_t)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, resp. <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g left-parenthesis upper X Subscript t Baseline right-parenthesis slash double-struck upper E g left-parenthesis upper X Subscript t Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>g</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">E</mml:mi>\n </mml:mrow>\n <mml:mi>g</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">g(X_t)/\\mathbb {E} g(X_t)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, are martingales.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-12-07","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/1157","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}

引用次数: 0

Abstract

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

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设X=(X t) t≥0 X=(X＿t){＿t\geq 0为}一维lsamvy过程，使得每个X t X＿t具有cb1 C^1＿b -密度w. r. t.勒贝格测度和某些多项式或指数矩。我们描述了所有多项式有界函数f: R→R f\colon\mathbb R{}\to\mathbb R{，以及指数有界函数g:R→(0，∞)g }\colon\mathbb R{}\to (0, \infty)，使得f(X t)−E f(X t) f(X＿t)- \mathbb E{ f(X＿t)g(X t)/ eg (X t) g(X＿t)/ }\mathbb eg (X＿t)是鞅。{}

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Theory of Probability and Mathematical Statistics STATISTICS & PROBABILITY-

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1.30

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