This paper does not contain anything new, and, in fact, it deals with a rather old-fashioned subject: the theorem of Chung-Doob [CD 64] and Meyer [Me66] (cf. also [DM 78]) on the existence of a progressively measurable modification of a measurable, adapted process. The raison d'€tre of this paper is that in our seminar at Mannheim periodically with new students the need comes up to understand the construction of the Itö-integral, and that in most of the common literat ure the details of a certain argument (see below) are not given or only hinted at. This paper attempts to provide that. argument in rather much - maybe even unnecessary - detail, so that the student can understand the probably most direct approach to the stochastic integral with respect to a Brownian motion as the limit of approximations of Riemann type.
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