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On the regularity of stochastic difference equations in hyperfinite- dimensional vector spaces and applications to \(\mathcal{D}'\)-valued stochastic differential equations. (English) Zbl 0819.60043

In the area of stochastic analysis, nonstandard analysis is proving to be a very significant tool. This paper is another example of how such procedures answer important questions in this area. The author uses nonstandard analysis in the sense of Robinson with respect to a polysaturated nonstandard model for a basic superstructure. The author gives a construction for a Wiener \({\mathcal D}'\)-process, \(W_ t\), \(t \in [0, \infty)\). Then a hyperfinite representation of stochastic integrals for operator-valued processes with respect to \(W_ t\) is derived. Existence theorems in the spirit of Keisler are proved for (infinite- dimensional) stochastic differential equations of Itô’s type one and a certain kind of Itô’s type two. This is done via a regularity of hyperfinite stochastic difference equations.

MSC:

60G20 Generalized stochastic processes
03H05 Nonstandard models in mathematics
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