Stochastic integration for abstract, two parameter stochastic processes I. Stochastic processes with finite semivariation in banach spaces
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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.
KeywordsBanach Space Additive Measure Integrable Variation Partial Function Stochastic Integration
- Brooks J. K., Dinculeanu N.,Stochastic Integration in Banach spaces, Seminar on Stochastic Process, 1990, Birkhäuser, (1991), 27–115.Google Scholar
- 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
- Lindsey C.,Two parameter Stochastic Processes with finite variation, PhD Thesis, Univ. of Florida (1988).Google Scholar