具有反射边界的摄动随机微分方程解的存在唯一性

IF 0.8 Q3 STATISTICS & PROBABILITY Monte Carlo Methods and Applications Pub Date : 2023-10-24 DOI:10.1515/mcma-2023-2018
Faiz Bahaj, Kamal Hiderah
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<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)&amp;=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)&amp;\\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 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<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)&amp;=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)&amp;\\\\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 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引用次数: 0

摘要

文摘本文在一些合适的条件下,我们证明强解的存在和唯一性的摄动随机微分方程反映边界(PSDERB) , { x⁢(t ) = x⁢(0)+∫0 tσ⁢(x⁢(年代 ) ) ⁢ d B⁢(s ) + ∫0 t b⁢(x⁢(年代 ) ) ⁢ d s +α⁢(t)⁢H⁢u (max 0≤≤t⁡x⁢(u ) ) + β⁢(t)⁢L t 0⁢(x)x⁢(t ) 所有⁢≥0 t≥0 , \ 左\{\{对齐}{}开始x (t)和= x (0) + \ int_ {0} ^ {t} \σ(年代,x (s)) \, dB (s) + \ int_ {0} ^ {t} b (s, x (s)) \ d + \α(t) H \ bigl {(} \ max_ {0 \ leq u \ leq t} x (u) \ bigr{)} + \β(t) L_ {t} ^ {0} (x) x (t)和\ \ \组0 \四\文本所有}{\ t \组0 \{对齐}\正确的结束。其中𝐻为连续的r值函数,σ,b, α \sigma,b, α \alpha,和时延为可测函数,L t 0 L_{t}^{0}表示半鞅变量的时间在零点处的局部时间。
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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.
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来源期刊
Monte Carlo Methods and Applications
Monte Carlo Methods and Applications STATISTICS & PROBABILITY-
CiteScore
1.20
自引率
22.20%
发文量
31
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