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Suppose I have a stochastic process $Z_t$ and a process $Y_t$ such that $$ dY_t = \mu(t,Z_t,Y_t)dt + \sigma(t,Z_t,Y_t)dW_t $$ and $W_t$ is independent of $Z_t$.

Then if I want to find dynamics for the conditional expectation $$ \mathbb{E}[ Y_t|\mathfrak{F}_t ], $$ where $\mathfrak{F}_t$ is the $\sigma$-field generated by $Z_t$, how can I do this (if we do not assume that $Y_t$ is somehow affine in $Z_t$)?

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if you assume that $\sigma$ is (appropriately) bounded such that the Ito integral is a true martingale (not only a local martingale), then the conditional expectation satisfies the following ODE $$ \frac{d \mathbb{E}[Y_t\mid \mathcal F_t]}{dt} = \mathbb{E}[\mu(t,Z_t,Y_t)\mid \mathcal{F}_t]. $$ Without having any further information on $\mu$ it is difficult to continue from here.

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