Computing Conditional Expectation with Continuous Martingales

Question

I am trying to compute the following question:

Let $(X_t,F_t)_{t \in \mathbb{R}}$ be a martingale with continuous realizations. For $0 \le s \le t$ find $E(\int_{0}^{t} X_u du | F_s).$

I am confused how to compute the conditional expectation with the integral inside.

Answer 1

First show that $E\int_0^t |X_u|du < \infty$, (see Exchange integral and conditional expectation for why). This is possible by Fubini and using that $E[|X_u|] \leq E[|X_t|]$ since $u \leq t$.

Then we have $$E[\int_0^t X_u\,du| F_s] = \int_0^t E[X_u|F_s]\,du = \int_0^s X_u\,du + X_s(t-s).$$