The Stochastic Fubini Theorem

Fubini’s theorem states that, subject to precise conditions, it is possible to switch the order of integration when computing double integrals. In the theory of stochastic calculus, we also encounter double integrals and would like to be able to commute their order. However, since these can involve stochastic integration rather than the usual deterministic case, the classical results are not always applicable. To help with such cases, we could do with a new stochastic version of Fubini’s theorem. Here, I will consider the situation where one integral is of the standard kind with respect to a finite measure, and the other is stochastic. To start, recall the classical Fubini theorem.

Theorem 1 (Fubini) Let {(E,\mathcal E,\mu)} and {(F,\mathcal F,\nu)} be finite measure spaces, and {f\colon E\times F\rightarrow{\mathbb R}} be a bounded {\mathcal E\otimes\mathcal F}-measurable function. Then,

\displaystyle  y\mapsto\int f(x,y)d\mu(x)

is {\mathcal F}-measurable,

\displaystyle  x\mapsto\int f(x,y)d\nu(y)

is {\mathcal E}-measurable, and,

\displaystyle  \int\int f(x,y)d\mu(x)d\nu(y)=\int\int f(x,y)d\nu(x)d\mu(y). (1)

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