7.8 White Noise Analysis

In this section, we use hypermodel to develope generalized Wiener functionals, in the sense of White Noise Analysis, can be developed. A full account of this will appear somewhere else. Some applications of Malliavin calculus in mathematical finance, such as those in 6.1.2, naturally produce terms that have no standard interpretation, but are objects in the white noise space. This space we construct is denoted by . It is large enough to include the Kontratiev space (S) ^?1, a well-known space in which White Noise Analysis can be developed. (See for example [Holden et al., 1996].) In fact is richer than (S) ^?1. For example, has a proper subspace called ₁ which contains all monomials and includes (S) ^?1 .

The space includes all L ²-Wiener functionals, all elements in the Krontratiev space, can do stochastic integration and stochastic differential equations, behaves well with respect to SL ²-liftings of Wiener functionals, behaves well with respect to composition with continuous functions (even with Schwartz distributions sometimes), is closed under the Wick product, contains interesting objects such as and Donsker s delta functions,

As before, let ? t:=1/ N, for some and use the hyperfinite timeline , n ²} and , equipped with an internal probability measure ? given by .