{"title":"具有不连续扩散系数的随机微分方程","authors":"Soledad Torres, Lauri Viitasaari","doi":"10.1090/tpms/1201","DOIUrl":null,"url":null,"abstract":"We study one-dimensional stochastic differential equations of the form <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d upper X Subscript t Baseline equals sigma left-parenthesis upper X Subscript t Baseline right-parenthesis d upper Y Subscript t\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dX_t = \\sigma (X_t)dY_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=\"application/x-tex\">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitable Hölder continuous driver such as the fractional Brownian motion <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Superscript upper H\"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">B^H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H greater-than one half\"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">H>\\frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which we assume very mild conditions. In particular, we allow <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stochastic differential equations with discontinuous diffusion coefficients\",\"authors\":\"Soledad Torres, Lauri Viitasaari\",\"doi\":\"10.1090/tpms/1201\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study one-dimensional stochastic differential equations of the form <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d upper X Subscript t Baseline equals sigma left-parenthesis upper X Subscript t Baseline right-parenthesis d upper Y Subscript t\\\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">dX_t = \\\\sigma (X_t)dY_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Y\\\"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitable Hölder continuous driver such as the fractional Brownian motion <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper B Superscript upper H\\\"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">B^H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H greater-than one half\\\"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">H>\\\\frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"sigma\\\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which we assume very mild conditions. In particular, we allow <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"sigma\\\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-10-03\",\"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/1201\",\"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/1201","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
摘要
我们研究了dX t = σ (X t)dY t dX＿t = \sigma (X＿t)dY＿t的一维随机微分方程，其中Y Y是一个合适的Hölder连续驱动器，如分数阶布朗运动B H B^H with H ＆gt;12 H＆gt;\frac本文的创新之处在于对扩散系数σ \sigma的假设，我们假设了非常温和的条件。特别地，我们允许σ \sigma具有不连续，因此我们的结果可以应用于研究具有不连续扩散的方程。
Stochastic differential equations with discontinuous diffusion coefficients
We study one-dimensional stochastic differential equations of the form dXt=σ(Xt)dYtdX_t = \sigma (X_t)dY_t, where YY is a suitable Hölder continuous driver such as the fractional Brownian motion BHB^H with H>12H>\frac 12. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients σ\sigma for which we assume very mild conditions. In particular, we allow σ\sigma to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.
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