The non-uniform Riemann approach to Itô's integral. (Q595804)

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.