LEADER 05861nam 22006495 450 001 9910254301203321 005 20220413214831.0 010 $a3-319-65678-3 024 7 $a10.1007/978-3-319-65678-6 035 $a(CKB)4100000000586902 035 $a(DE-He213)978-3-319-65678-6 035 $a(MiAaPQ)EBC5016870 035 $a(PPN)204534976 035 $a(EXLCZ)994100000000586902 100 $a20170901d2017 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aEquations involving Malliavin calculus operators $eapplications and numerical approximation /$fby Tijana Levajkovi?, Hermann Mena 205 $a1st ed. 2017. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2017. 215 $a1 online resource (X, 132 p. 7 illus., 6 illus. in color.) 225 1 $aSpringerBriefs in Mathematics,$x2191-8198 311 $a3-319-65677-5 320 $aIncludes bibliographical references. 327 $a1 White Noise Analysis and Chaos Expansions: 1.1 Introduction -- 1.3 Deterministic background -- 1.2 Spaces of random variables -- 1.4 Stochastic processes -- 1.5 Operators -- References -- 2 Generalized Operators of Malliavin Calculus: 2.1 Introduction -- 2.1 The Malliavin derivative -- 2.2 The Skorokhod integral -- 2.3 The Ornstein-Uhlenbeck operator -- 2.4 Properties of the Malliavin operators -- 2.5 Fractional operators of the Malliavin calculus -- References -- 3 Equations involving Mallivin Calculus Operators: 3.1 Introduction -- 3.2 Equations with the Ornstein-Uhlenbeck operator -- 3.3 First order equation with the Malliavin derivative operator -- 3.4 Nonhomogeneous equation with the Malliavin derivative operator -- 3.5 Wick-type equations involving the Malliavin derivative -- 3.6 Integral equation -- References -- 4 Applications and Numerical Approximation: 4.1 Introduction -- 4.1 A stochastic optimal control problem -- 4.3 Operator differential algebraic equations -- 4.4 Stationary equations -- 4.5 A fractional optimal control problem -- 4.6 Numerical approximation -- References. 330 $aThis book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed. The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator, the Skorokhod integral and the Ornstein-Uhlenbeck operator are introduced in terms of chaos expansions. The main properties of the operators, which are known in the literature for the square integrable processes, are proven using the chaos expansion approach and extended for generalized and test stochastic processes. Chapter 3, Equations involving Malliavin Calculus operators, is devoted to the study of several types of stochastic differential equations that involve the operators of Malliavin calculus, introduced in the previous chapter. Fractional versions of these operators are also discussed. Finally, in Chapter 4, Applications and Numerical Approximations are discussed. Specifically, we consider the stochastic linear quadratic optimal control problem with different forms of noise disturbances, operator differential algebraic equations arising in fluid dynamics, stationary equations and fractional versions of the equations studied ? applications never covered in the extant literature. Moreover, numerical validations of the method are provided for specific problems.". 410 0$aSpringerBriefs in Mathematics,$x2191-8198 606 $aProbabilities 606 $aFunctional analysis 606 $aPartial differential equations 606 $aCalculus of variations 606 $aNumerical analysis 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aCalculus of Variations and Optimal Control; Optimization$3https://scigraph.springernature.com/ontologies/product-market-codes/M26016 606 $aNumerical Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M14050 615 0$aProbabilities. 615 0$aFunctional analysis. 615 0$aPartial differential equations. 615 0$aCalculus of variations. 615 0$aNumerical analysis. 615 14$aProbability Theory and Stochastic Processes. 615 24$aFunctional Analysis. 615 24$aPartial Differential Equations. 615 24$aCalculus of Variations and Optimal Control; Optimization. 615 24$aNumerical Analysis. 676 $a519.2 700 $aLevajkovi?$b Tijana$4aut$4http://id.loc.gov/vocabulary/relators/aut$0766808 702 $aMena$b Hermann$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910254301203321 996 $aEquations Involving Malliavin Calculus Operators$92179912 997 $aUNINA