Moment Bounds on Ito Integrals

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.

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