On a Dual Pair of Spaces of Smooth and Generalized Random Variables

A dual pair G and G* of smooth and generalized random variables, respectively, over the white noise probability space is studied. G is constructed by norms involving exponentials of the Ornstein-Uhlenbeck operator, G* is its dual. Sufficient criteria are proved for when a function on S(IR) is the S-transform of an element in G or G*

Topological aspects of the characterization of Hida distributions – a remark

In the recent years, the dual pair of smooth and generalized random variables on the White Noise space, (S) and (S)*, has found many applications. For example, stochastic (partial) differential equations [L0U 90, L0U 91, Po 92, Po 93], quantum field theory [PS 93] and Feynman integrals [FPS 91, KS 92, LLS 93]. The main advantage of (S) and (S)* is the S-Transform, which in a nice way characterizes the pair. This transform maps generalized Hida distributions into a space of complex valued functions on S(IR). This space of functions is called the space of U-functionals. Moreover, the S-Transform turns out to be a bijection onto this space [PS 91]. In most applications, one is really working on the space of U-functionals. For this reason, it is natural to topologize the U-functional space. The aim of this paper is to construct the U-functional space using inductive and projective limits of Banach spaces. This construction is in light of the construction of (S) and (S)* quite natural. With the given topologies, we show our main result: The S-Transform is a homeomorphism

Some results about the large deviations principle in white noise analysis

In recent years, there has been an enormous interest in the theory of large deviations, i.e. in the asymptotic behaviour of small probabilities on an exponential scale. Although the roots of this theory can be dated back to Cramér in 1938 it took until the mid-1970's that starting with Donsker and Varadhan the subject exploded. Numbers of publications have been written since then, and the subject has found many applications to related fields like statistical mechanics or others. On the other hand white noise analysis provides lots of powerful tools as well for irrfinitedimensional calculus as for probability theory, a quite complete overview is given by. So the combination of these two subjects should inspire new results and give a feed-back to each of them. In the present paper I will do a very first step towards this aim stating some large deviations results in the context of white noise analysis. Not only the white noise probability measure μ will be considered, but also a certain dass of functionals over the white noise space (S' (IR), Β, μ) turns out to correspond to measures as first shown independently. For some of these measures large deviations results can be shown as well

Topological aspects of the characterization of Hida distributions – a remark

In the recent years, the dual pair of smooth and generalized random variables on the White Noise space, (S) and (S)*, has found many applications. For example, stochastic (partial) differential equations [L0U 90, L0U 91, Po 92, Po 93], quantum field theory [PS 93] and Feynman integrals [FPS 91, KS 92, LLS 93]. The main advantage of (S) and (S)* is the S-Transform, which in a nice way characterizes the pair. This transform maps generalized Hida distributions into a space of complex valued functions on S(IR). This space of functions is called the space of U-functionals. Moreover, the S-Transform turns out to be a bijection onto this space [PS 91]. In most applications, one is really working on the space of U-functionals. For this reason, it is natural to topologize the U-functional space. The aim of this paper is to construct the U-functional space using inductive and projective limits of Banach spaces. This construction is in light of the construction of (S) and (S)* quite natural. With the given topologies, we show our main result: The S-Transform is a homeomorphism

Die Evaluation des Risiko-Ertrags-Profils von Aktienpositionen mit Optionen auf der Grundlage des Shortfall-Ansatzes

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