On signed probability measures and some old results of Krylov

It is an interesting exercise to compute the iterated integrals of Brownian Motion and to calculate the expectations (of polynomial functions of these integrals).

Recent work on constructing discrete measures on path space, which give the same value as Wiener measure to certain of these expectations, has led to promising new numerical algorithms for solving 2nd order parabolic PDEs in moderate dimensions. Old work of Krylov associated finitely additive signed measures to certain constant coefficient PDEs of higher order. Recent work with Levin allows us to identify the relevant expectations of iterated integrals in this case, leaving many interesting open questions and possible numerical algorithms for solving high dimensional elliptic PDEs.