## 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.