Bound for the Brownian motion exit time

Bound for the Brownian motion exit time

Suppose T inft: B_tnotin a,b where a0b and aneq -b. I would like to show mathbbET2 leq C mathbbEB_T4. The problem also says to apply Cauchy-Schwarz inequality to mathbbETB_T2. Now I know B_t4 - 6tB_t2 3t2_t is a martingale, and it suffices to show the inequality above for Twedge t. By martingale property and Cauchy-Schwarz, we have mathbbE B_Twedge t4 3mathbbE Twedge t2 leq 6 biggmathbbE Twedge t2bigg12 bigg mathbbE B_Twedge t4bigg12 I am stuck here, and I am trying to conclude from this without getting into too explicit calculation with the B_T term since we can actually compute mathbbET2 and mathbbEB_T4, but I think it is not the point of this problem...

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