The Representation of Conditional Expectations for Non-gaussian Noise

Recently, the martingale property and conditional expectations w.r.t. the natural filtration of Brownian motion for (generalized) processes have been studied by [9], [3], [6], and [8] in the context of white noise analysis. For regular processes these characterizations are an immediate consequence of the chaos expansion w.r.t. multiple stochastic integrals. They have turned out to be useful for the study of local times, see [4] and the study of a generalized Clark-Ocone formula [1], [5], and [15]. This has motivated us to consider these features for a more general class of processes and more general systems of functions than multiple stochastic integrals. We shall work throughout with the space D(R) of generalized functions as our sample space; recall the Gelfand triple D(R) ⊂ L(R, dt) ⊂ D(R). One equips D(R) with the weak σ-algebra F(D(R)), i.e. the σ-algebra generated by the mappings ω 7→ 〈ω, φ〉 for φ ∈ D(R). A probability measure P on (D(R),F(D ′ (R))) gives rise to a generalized coordinate process Φ by Φ : D(R) ×D(R) → R; (φ, ω) 7→ 〈ω, φ〉 . (1.1)