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The non-uniform Riemann approach to Itô’s integral. (English) Zbl 1067.60025

This paper proposes two approaches to the Itô’s integral in the line of generalized Riemann integrals of the Kurzweil-Henstock-McShane type. It is proved that the two approaches provide a stochastic integral which is equivalent to Itô’s one. The definitions are based upon McShane belated divisions, both with variable meshes in the line of the classical Kurzweil-Henstock-McShane integrals. The first definition, motivated by the Henstock lemma for classical integration, makes use of a suitable concept of additive function of meshed intervals. The second definition is a slight modification of a definition of the stochastic integral given by McShane. The basic properties of the integral are proved.

MSC:

60H05 Stochastic integrals
26A39 Denjoy and Perron integrals, other special integrals
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