Stochastic integration for abstract,two parameter stochastic processes I. Stochastic processes with finite semivariation in banach spaces |
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Authors: | Nicolae Dinculeanu |
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Institution: | 1. Dept. of Mathematics, University of Florida, 32611 8105, Gainesville, FL, U.S.A.
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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. |
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