Advertisement

Stochastic integration for abstract, two parameter stochastic processes I. Stochastic processes with finite semivariation in banach spaces

  • Nicolae Dinculeanu
Article

Abstract

In this paper we define the stochastic integral for two parameter processes with values in a Banach spaceE. We use a measure theoretic approach. To each two parameter processX withX st L E p we associate a measureI X with values inL E p .

IfX isp-summable, i.e. ifI X can be extended to aσ-additive measure with finite semivariation on theσ-algebra of predictable sets, then the integralε HdI X can be defined and the stochastic integral is defined by (H·X) z =ε [0,z] HdI X .

We prove that the processes with finite variation and the processes with finite semivariation are summable and their stochastic integral can be computed pathwise, as a Stieltjes Integral of a special type.

Keywords

Banach Space Additive Measure Integrable Variation Partial Function Stochastic Integration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. [1]
    Brooks J. K., Dinculeanu N.,Stochastic Integration in Banach spaces, Seminar on Stochastic Process, 1990, Birkhäuser, (1991), 27–115.Google Scholar
  2. [2]
    Brooks J. K., Dinculeanu N.,Integration in Banach spaces. Application to Stochastic Integration, Atti Sem. Mat. Fis. Univ. Modena,43 (1995), 317–361.MathSciNetMATHGoogle Scholar
  3. [3]
    Cairoli R., Walsh J.,Stochastic Integrals in the plane, Acta Math.,134 (1975), 111–183.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Dellacherie C., Meyer P.,Probabilités et Potentiel, Hermann, Paris, 1975, 1980.MATHGoogle Scholar
  5. [5]
    Dinculeanu N.,Vector-valued Stochastic Processes I. Vector measures and vector-valued Stochactis Processes with finite variation, J. of Theoretical Prob.,1 (1988), 149–169.CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    Dinculeanu N.,Vector-valued Stochastic processes V. Optional and predictable variation of Stochastic measures and Stochastic processe, Proc. A.M.S.,104 (1988), 625–631.CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    Dinculeanu N.,Stochastic Processes with finite semivariation in Banach spaces and their Stochastic Integral, Rend. Circ. Mat. Palermo,48 (1999), 365–400.CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Dinculeanu N.,Stochastic Integration for abstract, two parameter stochastic processes II. Square integrable martingales in Hilbert spaces, Stochastic Analysis and Applications (to appear).Google Scholar
  9. [9]
    Kwapien S.,On Banach spaces containing c 0, Studia Math.,5 (1974), 187–188.MathSciNetGoogle Scholar
  10. [10]
    Lindsey C.,Two parameter Stochastic Processes with finite variation, PhD Thesis, Univ. of Florida (1988).Google Scholar
  11. [11]
    Meyer P. A.,Théorie élémentaire des processus à deux indices, Springer Lecture Notes in Math.,863 (1981), 1–39.CrossRefGoogle Scholar
  12. [12]
    Radu E.,Mesures Stieltjes vectorielles sur R n, Bull. Math. Soc. Sci. Math. R. S. Roumanie,9 (1965), 129–136.MathSciNetGoogle Scholar

Copyright information

© Springer 2000

Authors and Affiliations

  • Nicolae Dinculeanu
    • 1
  1. 1.Dept. of MathematicsUniversity of FloridaGainesvilleU.S.A.

Personalised recommendations