Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory

Let u(t, x), t ϵ R, be an adapted process parametrized by a variable x in some metric space X, μ(ω, dx) a probability kernel on the product of the probability space Ω and the Borel sets of X. We deal with the question whether the Stratonovich integral of u(., x) with respect to a Wiener process on Ω and the integral of u(t,.) with respect to the random measure μ(., dx) can be interchanged. This question arises, for example, in the context of stochastic differential equations. Here μ(., dx) may be a random Dirac measure δη(dx), where η appears as an anticipative initial condition. We give this random Fubini-type theorem a treatment which is mainly based on ample applications of the real variable continuity lemma of Garsia, Rodemich and Rumsey. As an application of the resulting “uniform Stratonovich calculus” we give a rigorous verification of the diagonalization algorithm of a linear system of stochastic differential equations.