Integral representations of solutions for linear stochastic equations with multiplicative perturbances
Identifieur interne : 000742 ( Istex/Corpus ); précédent : 000741; suivant : 000743Integral representations of solutions for linear stochastic equations with multiplicative perturbances
Auteurs : M. E. ShaikinSource :
- Automation and Remote Control [ 0005-1179 ] ; 2010-04-01.
Abstract
Abstract: We consider the problem of explicitly representing the solutions of multiplicatively perturbed stochastic equations. We represent the solution as an integral Cauchy formula whose transition matrix is random in the case of multiplicative perturbations. Similar to deterministic theory, the transition matrix can be expressed in terms of the fundamental matrix or given by a stochastic Peano series. We give equations for statistical moments of the state vector and explicit integral representations of their solutions. For computing transition matrices of equations on moments, we use some group-theoretical notions and results whose usefulness is illustrated with simple examples.
Url:
DOI: 10.1134/S0005117910040028
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We represent the solution as an integral Cauchy formula whose transition matrix is random in the case of multiplicative perturbations. Similar to deterministic theory, the transition matrix can be expressed in terms of the fundamental matrix or given by a stochastic Peano series. We give equations for statistical moments of the state vector and explicit integral representations of their solutions. For computing transition matrices of equations on moments, we use some group-theoretical notions and results whose usefulness is illustrated with simple examples.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>Original Russian Text © M.E. Shaikin, 2010, published in Avtomatika i Telemekhanika, 2010, No. 4, pp. 16–33.</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>The work was supported by the Russian Foundation for Basic Research, project no. 09-08-00595-a and the Presidium of Russian Academy of Sciences (Program 29).</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Integral representations of solutions for linear stochastic equations with multiplicative perturbances</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Integral representations of solutions for linear stochastic equations with multiplicative perturbances</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">M.</namePart><namePart type="given">E.</namePart><namePart type="family">Shaikin</namePart><affiliation>Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>SP MAIK Nauka/Interperiodica</publisher><place><placeTerm type="text">Dordrecht</placeTerm></place><dateCreated encoding="w3cdtf">2009-03-02</dateCreated><dateIssued encoding="w3cdtf">2010-04-01</dateIssued><dateIssued encoding="w3cdtf">2010</dateIssued><copyrightDate encoding="w3cdtf">2010</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: We consider the problem of explicitly representing the solutions of multiplicatively perturbed stochastic equations. We represent the solution as an integral Cauchy formula whose transition matrix is random in the case of multiplicative perturbations. Similar to deterministic theory, the transition matrix can be expressed in terms of the fundamental matrix or given by a stochastic Peano series. We give equations for statistical moments of the state vector and explicit integral representations of their solutions. 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