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{"title":"具有反射边界的摄动随机微分方程解的存在唯一性","authors":"Faiz Bahaj, Kamal Hiderah","doi":"10.1515/mcma-2023-2018","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, under some suitable conditions, we prove existence of a strong solution and uniqueness for the perturbed stochastic differential equations with reflected boundary (PSDERB), that is, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing=\"0pt\" displaystyle=\"true\" rowspacing=\"0pt\"> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"0.055em\">+</m:mo> <m:mrow> <m:msubsup> <m:mo>∫</m:mo> <m:mn>0</m:mn> <m:mi>t</m:mi> </m:msubsup> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo lspace=\"0.170em\"></m:mo> <m:mrow> <m:mo mathvariant=\"italic\" rspace=\"0em\">d</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace=\"0.055em\">+</m:mo> <m:mrow> <m:msubsup> <m:mo>∫</m:mo> <m:mn>0</m:mn> <m:mi>t</m:mi> </m:msubsup> <m:mrow> <m:mi>b</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo lspace=\"0.170em\"></m:mo> <m:mrow> <m:mo mathvariant=\"italic\" rspace=\"0em\">d</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>H</m:mi> <m:mo></m:mo> <m:mrow> <m:mo maxsize=\"120%\" minsize=\"120%\">(</m:mo> <m:mrow> <m:mrow> <m:munder> <m:mi>max</m:mi> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>u</m:mi> <m:mo>≤</m:mo> <m:mi>t</m:mi> </m:mrow> </m:munder> <m:mo lspace=\"0.167em\"></m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo maxsize=\"120%\" minsize=\"120%\">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>t</m:mi> <m:mn>0</m:mn> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mrow> <m:mrow> <m:mi /> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mspace width=\"1em\" /> <m:mrow> <m:mrow> <m:mtext>for all</m:mtext> <m:mo lspace=\"0.500em\"></m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \\left\\{\\begin{aligned} {}x(t)&=x(0)+\\int_{0}^{t}\\sigma(s,x(s))\\,dB(s)+\\int_{0}^{t}b(s,x(s))\\,ds+\\alpha(t)H\\bigl{(}\\max_{0\\leq u\\leq t}x(u)\\bigr{)}+\\beta(t)L_{t}^{0}(x),\\\\ x(t)&\\geq 0\\quad\\text{for all}\\ t\\geq 0,\\end{aligned}\\right. where 𝐻 is a continuous R-valued function, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>σ</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:math> \\sigma,b,\\alpha and 𝛽 are measurable functions, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>L</m:mi> <m:mi>t</m:mi> <m:mn>0</m:mn> </m:msubsup> </m:math> L_{t}^{0} denotes a local time at point zero for the time of the semi-martingale 𝑥.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"36 1-2","pages":"0"},"PeriodicalIF":0.8000,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence and uniqueness of solutions for perturbed stochastic differential equations with reflected boundary\",\"authors\":\"Faiz Bahaj, Kamal Hiderah\",\"doi\":\"10.1515/mcma-2023-2018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this paper, under some suitable conditions, we prove existence of a strong solution and uniqueness for the perturbed stochastic differential equations with reflected boundary (PSDERB), that is, <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing=\\\"0pt\\\" displaystyle=\\\"true\\\" rowspacing=\\\"0pt\\\"> <m:mtr> <m:mtd columnalign=\\\"right\\\"> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\\\"left\\\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\\\"0.055em\\\">+</m:mo> <m:mrow> <m:msubsup> <m:mo>∫</m:mo> <m:mn>0</m:mn> <m:mi>t</m:mi> </m:msubsup> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo lspace=\\\"0.170em\\\"></m:mo> <m:mrow> <m:mo mathvariant=\\\"italic\\\" rspace=\\\"0em\\\">d</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace=\\\"0.055em\\\">+</m:mo> <m:mrow> <m:msubsup> <m:mo>∫</m:mo> <m:mn>0</m:mn> <m:mi>t</m:mi> </m:msubsup> <m:mrow> <m:mi>b</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo lspace=\\\"0.170em\\\"></m:mo> <m:mrow> <m:mo mathvariant=\\\"italic\\\" rspace=\\\"0em\\\">d</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>H</m:mi> <m:mo></m:mo> <m:mrow> <m:mo maxsize=\\\"120%\\\" minsize=\\\"120%\\\">(</m:mo> <m:mrow> <m:mrow> <m:munder> <m:mi>max</m:mi> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>u</m:mi> <m:mo>≤</m:mo> <m:mi>t</m:mi> </m:mrow> </m:munder> <m:mo lspace=\\\"0.167em\\\"></m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo maxsize=\\\"120%\\\" minsize=\\\"120%\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>t</m:mi> <m:mn>0</m:mn> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\\\"right\\\"> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\\\"left\\\"> <m:mrow> <m:mrow> <m:mrow> <m:mi /> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mspace width=\\\"1em\\\" /> <m:mrow> <m:mrow> <m:mtext>for all</m:mtext> <m:mo lspace=\\\"0.500em\\\"></m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \\\\left\\\\{\\\\begin{aligned} {}x(t)&=x(0)+\\\\int_{0}^{t}\\\\sigma(s,x(s))\\\\,dB(s)+\\\\int_{0}^{t}b(s,x(s))\\\\,ds+\\\\alpha(t)H\\\\bigl{(}\\\\max_{0\\\\leq u\\\\leq t}x(u)\\\\bigr{)}+\\\\beta(t)L_{t}^{0}(x),\\\\\\\\ x(t)&\\\\geq 0\\\\quad\\\\text{for all}\\\\ t\\\\geq 0,\\\\end{aligned}\\\\right. where 𝐻 is a continuous R-valued function, <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>σ</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:math> \\\\sigma,b,\\\\alpha and 𝛽 are measurable functions, <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msubsup> <m:mi>L</m:mi> <m:mi>t</m:mi> <m:mn>0</m:mn> </m:msubsup> </m:math> L_{t}^{0} denotes a local time at point zero for the time of the semi-martingale 𝑥.\",\"PeriodicalId\":46576,\"journal\":{\"name\":\"Monte Carlo Methods and Applications\",\"volume\":\"36 1-2\",\"pages\":\"0\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-10-24\",\"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-2023-2018\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monte Carlo Methods and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/mcma-2023-2018","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
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Existence and uniqueness of solutions for perturbed stochastic differential equations with reflected boundary
Abstract In this paper, under some suitable conditions, we prove existence of a strong solution and uniqueness for the perturbed stochastic differential equations with reflected boundary (PSDERB), that is, { x ( t ) = x ( 0 ) + ∫ 0 t σ ( s , x ( s ) ) d B ( s ) + ∫ 0 t b ( s , x ( s ) ) d s + α ( t ) H ( max 0 ≤ u ≤ t x ( u ) ) + β ( t ) L t 0 ( x ) , x ( t ) ≥ 0 for all t ≥ 0 , \left\{\begin{aligned} {}x(t)&=x(0)+\int_{0}^{t}\sigma(s,x(s))\,dB(s)+\int_{0}^{t}b(s,x(s))\,ds+\alpha(t)H\bigl{(}\max_{0\leq u\leq t}x(u)\bigr{)}+\beta(t)L_{t}^{0}(x),\\ x(t)&\geq 0\quad\text{for all}\ t\geq 0,\end{aligned}\right. where 𝐻 is a continuous R-valued function, σ , b , α \sigma,b,\alpha and 𝛽 are measurable functions, L t 0 L_{t}^{0} denotes a local time at point zero for the time of the semi-martingale 𝑥.