For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as:

$$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$

for $t\leq u$. I have a few questions about properties of $M_t$:

- Is it a uniformly integrable martingale? (or under what conditions it is)
- According to the martingale representation theorem it can be written as:

$$M_t(u) = M_0(u) + \int_0^t v_s(u) M_s(u) dW$$

where $W$ is the Brownian motion generating the filtration. What are the properties of $v_s(u)$? Does it need to be square-integrable? Bounded?

EDIT: I assume standard Brownian filtration. $u$ is regarded as a parameter, so I am only interested in behaviour in $t$.

`$a_u$`

is integrable), because`$a_u$`

is just a fixed random variable an you can use Jensen for conditional probabilities. $\endgroup$ – Wolfgang Loehr Sep 10 '12 at 12:46`$a_t$`

is a continuous process? $\endgroup$ – Wolfgang Loehr Sep 10 '12 at 15:19