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

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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 p E we associate a measureI X with values inL p E .

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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  1. Dept. of Mathematics, University of Florida, 32611 8105, Gainesville, FL, U.S.A.

    Nicolae Dinculeanu

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  1. Nicolae Dinculeanu
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Dinculeanu, N. Stochastic integration for abstract, two parameter stochastic processes I. Stochastic processes with finite semivariation in banach spaces. Rend. Circ. Mat. Palermo 49, 259–306 (2000). https://doi.org/10.1007/BF02904234

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  • DOI: https://doi.org/10.1007/BF02904234

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