Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle M \right \rangle_\infty \leq 1$ $\mathbb P$-ps. Now, we define by recurence $ \forall n \in \mathbb{N}$ $$ I^{(o)}_t \equiv 1, \ I^{(n+1)}_t \int _0 ^t I^{(n)}_s d M_s \ , \ t \geq 0 $$
The question: How to show that $I^{(n)}$ is a $\mathcal F _t $-martingal with $I^{(n)}_t$ in $L^2(\Omega, \mathcal F, \mathbb P)$ and that we have
$$ \mathbb{E} \left \{ \sup_ {t \geq 0} \left | I^{(n)}_t \right | ^2 \right \} \leq 4^n $$
Element of answer: for the last inequality we can maybe start using Burkholder-Davis-Gundy inequality, then we'll have :
\begin{align} \mathbb{E} \left \{ \sup_ {0 \leq t \leq T} \left | I^{(n)}_t \right | ^2 \right \} & \overset{BDG}{\leq} C\mathbb{E} \left \{ \left \langle I^{(n)} \right \rangle_T \right \} \\&= C\mathbb{E} \left \{ \int_0 ^T \left |I^{(n-1)}_s\right | ^2 d\left \langle M\right \rangle _s \right \} \\ & \overset{Fubini}{=} C \int_0 ^T \mathbb{E} \left \{\left |I^{(n-1)}_s\right | ^2 \right \}d\left \langle M\right \rangle _s \\ & \overset{Ito's isometry}{=} C \int_0 ^T \mathbb{E} \left \{\int _0 ^s\left |I^{(n-2)}_s\right | ^2 d\left \langle M\right \rangle _{s_1}\right \}d\left \langle M\right \rangle _s \\ &=(...) \\& = C \int_0 ^T \int_0 ^s \int_0 ^{s_1} ... \int_0 ^{s_n-2} 1 \ d\left \langle M\right \rangle _{s_{n-2}} ... d\left \langle M\right \rangle _{s_2} d\left \langle M\right \rangle _{s_1} d\left \langle M\right \rangle _{s}\end{align}
EDIT
Did sugestion: Could you please, check if I understood ?
You should notice that all the steps where the Fubini's theorem was applied are wrong!
\begin{align} \mathbb{E} \left \{ \sup_ {0 \leq t \leq T} \left | I^{(n)}_t \right | ^2 \right \} & \overset{BDG}{\leq} 4\mathbb{E} \left \{ \left \langle I^{(n)} \right \rangle_T \right \} \\&=4\mathbb{E} \left \{ \int_0 ^T \left |I^{(n-1)}_s\right | ^2 d\left \langle M\right \rangle _s \right \} \\ & \overset{Fubini}{=} 4 \int_0 ^T \mathbb{E} \left \{\left |I^{(n-1)}_s\right | ^2 \right \}d\left \langle M\right \rangle _s \\ & \overset{}{\leq} 4 \int_0 ^T \mathbb{E} \left \{\sup_{0\leq u \leq s}\left |I^{(n-1)}_u\right | ^2 \right \}d\left \langle M\right \rangle _s \\ & \overset{BDG}{=} 4 ^2 \int_0 ^T \mathbb{E} \left \{\int _0 ^s\left |I^{(n-2)}_s\right | ^2 d\left \langle M\right \rangle _{s_1}\right \}d\left \langle M\right \rangle _s \\ &=(...) \\& \leq 4^n \int_0 ^T \int_0 ^s \int_0 ^{s_1} ... \int_0 ^{s_n-2} 1 \ d\left \langle M\right \rangle _{s_{n-2}} ... d\left \langle M\right \rangle _{s_2} d\left \langle M\right \rangle _{s_1} d\left \langle M\right \rangle _{s}\\ &\leq 4^n\end{align} because $\left \langle M \right \rangle = (\left \langle M \right \rangle_t)_{t\geq 0}$ is an increasing process with $\left \langle M \right \rangle_\infty \leq 1$
