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I have a process $f_t$ adapted to a Weiner process, $W_t$. Moreover, $f_t$ is uniformly bounded by a constant, i.e., $|f_t| \leq C$. How can I bound

$$ \mathbb E \left[ \exp \left( \sup_{0 < u < t} \left | \int_0^u f_s \ d W_s \right| \right) \right] $$

I thought that I could use

$$ \left |\int_0^t f_s \ d W_s \right| \leq C \left|\int_0^t \ d |W|_s \right| = C \ |W_t | $$

However, I cannot quite prove the inequality. In fact, I am not sure it's true.

Tohiko
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    That inequality is not true. Here are some hints: $\int_0^t f_s dW_s$ is a time changed Brownian motion by the Dubins-Schwarz theorem and the supremum of Brownian motion on an interval has sub-Gaussian tails by the reflection principle. – Chris Janjigian Jun 27 '18 at 12:23
  • Thanks for the great hint! Using this I managed to bound the expectation. – Tohiko Jun 27 '18 at 13:44
  • Coming back to this question, I am note sure how I would use the reflection principle here, since the I have $\sup_{0 < u < t} \left|W_{\langle M \rangle_t} \right| $ for $M_t = \int_0^t f_s dW_s$. See here for a question on this: – Tohiko Jul 30 '18 at 16:17
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    As a secondary hint here, you will need to use the fact that $\langle M \rangle_t \leq C^2 t$. I have added a hint to your other question as well. – Chris Janjigian Jul 30 '18 at 22:33

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