It is a classical result that if $X$ is a process with values in $\mathbb{R}^d$ and \begin{align*} M_t^i = X_t^i - \int_0^t b_i(X_s) ds \\ M_t^i M_t^j - \int_0^t a_{ij} (X_s) ds \end{align*} are both (local) martingales then (on an enlargement of the underlying probability space) $X$ is a solution of the SDE $dX_t = b(X_t) dt + \sigma(X_t) dB_t$ for some Brownian motion $B_t$ where $a = \sigma \sigma^T$.
In the paper "Genealogies in expanding populations" by Durrett and Fan it is claimed that if for any $\phi, \psi \in C_c^{1,2}([0, \infty) \times \mathbb{R})$ we have that \begin{align*} \langle u_t, \phi_t \rangle - \langle u_0, \phi_0 \rangle + \langle l_t , \psi_t \rangle - \langle l_0, \psi_0 \rangle - \alpha \int_0^t \langle u_s, \partial_s \phi_s + \Delta \phi_s \rangle + \langle l_s, \partial_s \psi_s + \Delta \psi_s \rangle ds \\- 2 \theta \beta \int_0^t \langle u_s(1-u_s), \phi_s \rangle + \langle l_s(1-l_s), \psi_s \rangle ds \end{align*} is a continuous martingale with quadratic variation $$4 \gamma \int_0^t \langle u_s(1-u_s), \phi_s^2 \rangle + \langle l_s(1-l_s) \psi^2 \rangle + 2 \langle l_s(1-u_s), \phi_s \psi_s \rangle ds$$ then we can find independent space-time white noises $W^0, W^1, W^2$ such that
\begin{align*} \partial_t u &= \alpha \Delta u + 2 \theta \beta u(1-u) + \sqrt{4 \gamma l(1-u)} \dot{W^0} + \sqrt{4\gamma (u-l)(1-u)}\dot{W^1} \\ \partial_t l &= \alpha \Delta l + 2 \theta \beta l(1-u) + \sqrt{4 \gamma l(1-u)} \dot{W^0} + \sqrt{4\gamma l(u-l)}\dot{W^2} \end{align*} in the weak sense. They claim that this result is standard and refer to a similar result relating solutions of a martingale problem to those of a single SPDE against a single white noise in a paper of Mueller and Tribe. Mueller and Tribe also don't prove this result and refer instead to a proof of the SDE case. I am fairly sure that I have managed to prove the result of Mueller and Tribe relating solutions of suitable martingale problems to those of SPDEs against space-time white noise by adapting the standard SDE proof.
However I am unable to prove this result of Durrett and Fan. (The coupled system and the presence of multiple white noises cause my original approach trouble as far as I can tell.) I was hoping that someone would be able to point me towards a proof of the result of Durrett and Fan (or something similar). Failing this, is there somewhere I can find a generalisation of the SDE case to a coupled system with multiple Brownian motions so that I might attempt to adapt the proof of that result?
