The representation of conditional expectations for non-Gaussian noise

Abstract

For Wiener spaces conditional expectations and L 2 -martingales w.r.t. the natural flltration have a natural representation in terms of chaos expansion. In this note an extension to larger classes of processes is discussed. In particular, it is pointed out that orthogonality of the chaos expansion is not required. Recently, the martingale property and conditional expectations w.r.t. the nat- ural flltration 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 0 (R) of generalized functions as our sample space; recall the Gelfand triple D(R) ‰ L 2 (R;dt) ‰ D 0 (R). One equips D 0 (R) with the weakae-algebra F(D 0 (R)), i.e. theae-algebra generated by the map- pings ! 7! h!;'i for ' 2 D(R). A probability measure P on (D 0 (R);F(D 0 (R))) gives rise to a generalized coordinate process ' by