I'm interested in the following problem because of its relevance to the pricing of Asian-type options when the average is arithmetic and in continuous time:
If $(X_t,\mathcal{F}_t)_{0\leq t\leq T}$ is a martingale with $\mathrm{E}[X_t]=\mu$, and $Y=(Y_t)$ is the process defined by $Y_t=\int_{0}^{t}X_u\mathrm{d}u$, is it possible to find an expression for the conditional expectation $\mathrm{E}[Y_t|\mathcal{F}_s]$, $s\leq t$, that incorporates the information that $\mathrm{E}[X_t]=\mu$?
I'm not sure how to proceed, beyond unravelling the basic definitions:
$\mathrm{E}[\int_{0}^{t}X_u\mathrm{d}u|\mathcal{F}_s] = \mathrm{E}[\int_{0}^{s}X_u\mathrm{d}u|\mathcal{F}_s] + \mathrm{E}[\int_{s}^{t}X_u\mathrm{d}u|\mathcal{F}_s]$
I believe that the first term on the right-hand side is $\int_{0}^{s}X_u\mathrm{d}u$ because $Y_s$ is $\mathcal{F}_s$-measurable and the second term is equal to
$\int_{s}^{t}\mathrm{E}[X_u|\mathcal{F}_s]\mathrm{d}u$
(using the defining property of conditional expectation, sometimes called "partial averaging property") which, in turn, is equal to
$\int_{s}^{t}X_s\mathrm{d}u = X_s\cdot(t-s)$
because $X$ is a martingale. However, the information on the expected value of $X_t$ is left unused.