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I'm trying to derive the estimate $$ E\left[\left|\int_{0}^{t}h_r\,dB_r\right|^4\right] \leq 3C^4t^2,$$ where $h_r$ is continuous, adapted (to the natural Brownian filtration up to time $t$) and bounded (with constant $C$).

I almost got this estimate, except that I have no idea how to infer that the expectation $$ E\left[\int_{0}^{t}X_r^3 h_r\, dB_r\right]$$ is zero (if it were always negative, it would suffice as well, but I highly doubt this is the case here). To get this far I applied the Ito's formula to $dX_t=h_tdB_t$ and $f(x)=x^4$.

KKK
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    Use that stochastic integrals (with respect to Brownian motion) are (local) martingales. – saz Apr 16 '16 at 05:51
  • @saz Thank you, but I'm not supposed to use any martingale theory for this exercise. Even if I could I don't know much about martingales yet to use your hint. – KKK Apr 16 '16 at 12:15
  • You don't have to know much about martingales; just that martingales have constant expectation. – saz Apr 16 '16 at 13:04

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I think you can use the Wiener integral to solve that problem ?

$\int_{0}^{t}h_rdB_r$~ $ \mathscr{N}(0,\int_{0}^{t}h_r^2dr)$ as h is continuous adapted to the filtration (Wiener integral)

Then using the 4th moment of a central normal distribution,

$E[\int_{0}^{t}h_rdB_r] = 3*(\int_{0}^{t}h_r^2dr)^2 < 3C^4t^2$ using the fact that h is bounded.

ashu24
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  • I understand that $E[\int_0^t h_rdB_r]=0$ and that the variance follows from the Ito's isometry, but why does it mean that the integral is normally distributed? – KKK Apr 16 '16 at 12:19
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    @Kamil, hi Kamil, you can actually show this : $\int_{0}^{t}f(r)dB_r $~ $ B_\int_{0}^{t}f(r)^2dr $ for f in $L^2$ by the following step. First you consider f a step function and you show that $\int_{0}^{t}f(r)dB_r$ is a normal distribution ( using the independency of the brownian increments, then for f $in L^2$ you can approximate this function by a step function in order to have your result, using this trick for a continuous function allows you to show that your integral follows a normal distribution. – ashu24 Apr 16 '16 at 12:23
  • Thank you, I worked out the details :) – KKK Apr 16 '16 at 12:40