An Andô-Douglas type theorem in Riesz spaces with a conditional expectation

Abstract

In this paper we formulate and prove analogues of the Hahn-Jordan decomposition and an Andô-Douglas-Radon-Nikodým theorem in Dedekind complete Riesz spaces with a weak order unit, in the presence of a Riesz space conditional expectation operator. As a consequence we can characterize those subspaces of the Riesz space which are ranges of conditional expectation operators commuting with the given conditional expectation operators and which have a larger range space. This provides the first step towards a formulation of Markov processes on Riesz spaces. Mathematics Subject Classification (2000). 47B60, 60G40, 60G48, 60G42.

Doob’s optional sampling theorem in Riesz spaces

“…We need the following existence theorem and therefore have to prove Watson's Radon-Nikodym theorem for the more general case: Theorem 3.3 Let E be an F-universally complete Riesz space with conditional expectation operator F. Let F be a Dedekind complete Riesz subspace of E with R(F) ⊂ F. For each X ∈ E + there exists a unique Y ∈ F + such that FPX = FPY, for all P ∈ P F . Proof The uniqueness part follows exactly as in [24]. We now prove the existence.…”

“…We shall denote the set of all order projections of E that maps F into itself by P F and this set can be identified with the set of all order projections of the vector lattice F (we say these projections act on F, see [5]). The next notion, due to Watson [24], is now defined in this more general case.…”

Discrete stochastic integration in Riesz spaces

“…It should be noted thatT i could equally have been constructed as the unique conditional expectation havingT i T 1 [25].…”

“…. , f n } [25]. Here f := ( f n , T n ) is a martingale from Mart 2 (T n ) [22], and stochastic integrals with respect to f can be computed.…”

Andô–Douglas type characterization of optional projections and predictable projections

“…[19] further investigated the conditional expectations on Riesz spaces; in particular, it studied the extension of conditional expectations to their domains in the operator theoretic setting. Recently, [26] generalized the Andô-Douglas theorem to the Riesz spaces; the key results there include the Andô-Douglas characterization of generalized conditional expectation (cf. Section I.4 of [16]) as a special case.…”