Classical stochastic Fubini theorems start from a fixed semimartingale S, say, and a family of integrands for S which are parametrized by a parameter z, say, from some parameter space Z. There is a (non- random) measure μ, say, on Z, and the stochastic Fubini theorem then says that integration with respect to μ and stochastic integration with respect to S can be interchanged. In other words, we can either 1) first integrate the parametrized integrands with respect to μ and then stochastically integrate the resulting process with respect to S, or we can 2) first stochastically integrate, for each z, the corresponding integrand with respect to S and then integrate the result with respect to μ – and both double integrals yield the same result. What happens now if we replace the fixed measure μ by a stochastic kernel from the predictable sigma- field to Z? Approach 1) still makes sense, but how about 2)? And can we still get some kind of stochastic Fubini theorem? We show that we can, but we need to define for that a stochastic integral, with respect to S, of suitable measure-valued processes. The origin of this question comes from a (still open) question in mathematical finance. There are also some connections to a class of Volterra-type semimartingales. Based on joint work with Tahir Choulli and Martin Schweizer.
Jiaming Chen is a first year PhD student in mathematics at NYU Courant. He did his MS thesis at ETH Zurich.