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

Abstract

Let u( t, x), t ϵ R, be an adapted process parametrized by a variable x in some metric space X, μ( ω, d x) 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(., d x) can be interchanged. This question arises, for example, in the context of stochastic differential equations. Here μ(., d x) may be a random Dirac measure δ η (d x), 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.