In "Mitter SK, Newton NJ. A Variational Approach to Nonlinear Estimation. SIAM J Control Optim. 2003 Jan;42(5):1813–33", it is shown that the path estimation measure $P_{X|Y}(\cdot,y)$ for stochastic system $$dX_t=f(X_t)dt+g(X_t)dW_t,$$ $$dY_t=h(X_t)dt+dV_t,$$
can be represented by a controlled diffusion process
$$d\tilde X_t=\Big(f(\tilde X_t)+g(\tilde X_t)^2u(\tilde X_t,t,y)\Big)dt+g(\tilde X_t)d\tilde W_t,$$
where for each observation path $y$, $u(x,t,y)=\partial_x v(x,t,y)$, where $v$ is the solution to the backward stochastic partial differential equation
$$dv=-\mathcal{A}v dt- h(x)v dY_t, \quad v(x,T,y)=1,$$
called the Pardoux equation. Here, $\mathcal A$ denotes the infinitesimal generator of $X$.
I would like to know whether there is a similar representation for the filtering distribution $\pi_t(\cdot, y^t)$. There is a short remark about something along these lines in the above-mentioned paper, so it might be that this is already reported in the literature.