On a certain martingale representation and the related infinite dimensional moment problem

Abstract

It is well-known that any $L^2$-martingale with respect to a Brownian filtration is represented by a stochastic integral with respect to the Brownian motion. The theorem can be proven based on the fact that linear combinations of exponential martingales (of a specific type) are dense in the mentioned set. In this paper, the necessary and sufficient conditions for expressing martingales as true identities rather than approximations are considered, which turns out to be an infinite dimensional moment problem. A typical moment problem is given as follows: for real sequences $(\mu _i)_{i=0}^\infty$, find the necessary and sufficient conditions for the existence of a distribution whose support is a subset of $[0,\infty )$ and the $i$-th moments is $\mu _i$. This is a fundamental problem in probability theory or integral theory that was first proposed around 1894, but it is still being studied as of 2023. In this paper, we point out that this problem is related to the above problem through chaos expansion, and give a proof using a version of the moment problem.