A simple question:
Let $H$ be a cadlag, adapted process and $A$ a process of finite variation. Then also
$\int_t^T HdA_t$ is a finite variation process (see "Limit Theorems... "Jacod&Shiryaev Prop. 3.5 p. 28). Is $\mathbb{E}\left[\int_t^T HdA_t\right]$ also a finite variation process?
I think one can refute that:
Suppose $H\equiv1$. Then $\mathbb{E}\left[\int_t^T HdA_t\right]=\mathbb{E}\left[A_T-A_t\right]=\mathbb{E}\left[A_T\right]-A_t$
$\mathbb{E}[A_T|\mathcal{F}_t]$ is a true martingale, if $A_T$ is integrable. Now if $\mathbb{E}[A_T|\mathcal{F}_t]$ is predictable (esp. if it is continuous) then it can be of finite variation only, if it is constant. In other words, $A$ must be deterministic, if $\mathbb{E}[A_T|\mathcal{F}_t]$ is of finite variation.
Do you agree?