The Setup
---------


Let $\xi\_t$ be a process adapted to the filtration $\mathfrak{F\_t}$ of the semi-martinagale $X\_t$, such that both are square integrable. Then is the map
\begin{align}
F\_T: L^2(\mathfrak{F\_t},\mathbb{P}\times m) \rightarrow & L^2(\Omega,\mathbb{P}),\\
\xi\_t \mapsto & \int\_0^T \xi\_tdX\_t
\end{align}
where $m$ is the Lebesgue measure and $(\Omega,\mathfrak{F},\mathfrak{F}\_t,\mathbb{P})$ is a stochastic base for the process $X\_t$. 


The Question
------------


For any given $T>0$ is the map $F\_T$ a continuous operator, as it is clearly $\mathbb{R}$-linear? If not is it at least closed or bounded?


*Note:* Ultimately, my goal is to obtain some sort of boundedness result. 



Stochastic integral is a continous or closed operator?
You want an inequality like $E(\int \xi\_t dX\_i)^2 < cE\int \xi^2\_t dt$ (this is part question as I am not sure what the norm on the rhs is ), however, if $X\_t = \int \sigma(t) dW\_t$ where $\sigma $ is deterministic and W Brownian motion you get $E(\int \xi\_t dX\_i)^2 = E\int \xi^2\_t \sigma^2(t) dt$ . As $X$ is fine as long as $\int \sigma^2(t) dt < \infty$ but $\sigma $ does not have to be bounded the map will not be bounded. 


