Use Ito’s formula to show that if \(\sigma(t,\omega)\) is a
nonanticipating random function which is bounded. That is to say

\[ |\sigma(t,\omega)|\leq M\]

for all \(t \geq 0\) and all \(\omega\).

1. Under this assumption show that the stochastic integral
   \[I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)\]
   satisfies  the following moment estimates
   \[\mathbf E\{ |I(t)|^{2p}\}\leq 1\cdot 3\cdot 5 \cdots (2p-1) (M^2 t)^p\]
   for \(p=1,2,3,…\) if one assumes
   \[ \mathbf E \int_0^t |I(s)|^k \sigma(s) dB(s) =0\]
   for any integer \(k\).
2. Prove the above result without assuming that
   \[ \mathbf E \int_0^t |I(s)|^k \sigma(s) dB(s) =0\]
   since this requires that
   \[ \mathbf E \int_0^t |I(s)|^{2k} \sigma^2(s) ds  < \infty\]
   which we do not know a priory.