# derivative of conditional expectation operators $\mathbb{E}_t$?

Let $(\mathcal{F}_t)_{t\in [0,T]}$ be a filtration on a probability space $\Omega$. Fix $1<p<\infty$. Let $\{\mathcal{E}(\cdot|\mathcal{F}_t)\ :\ t\in [0,T] \}$ the associated family of conditional expectation operators on $L^p(\Omega)$. Fix $f\in L^p(\Omega)$. Consider the map $$t\mapsto \mathcal{E}(\cdot|\mathcal{F}_t)f.$$

I am interesting in information on the derivative of this map (possibly with stronger assumptions). Do you know references or results in this area?

remark: I know nothing information on the existence of this derivative.

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when $F_t$ is the brownian filtration $\mathbb E( X \vert F_t)$ is a stochastic integral, and can be represented as $\int^t \pi_s dW_s$. Finding the $\pi$ is the topic of the clark -ocone formula. – mike Sep 8 '12 at 19:13