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 \(\omega\).

- 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 \(k\). - 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.