Let \(\sigma(t,\omega)\) be nonanticipating with \(|\sigma(x,\omega)| < M\) for some bound \(M\) . Let \(I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)\). Use the exponential martingale \[\exp\big\{\alpha I(t)-\frac{\alpha^2}{2}\int_0^t \sigma^2(s)ds \big\}\] (see the problem here) and the Kolmogorov-Doob inequality to get the estimate

\[

P\Big\{ \sup_{0\leq t\leq T}|I(t)| \geq \lambda \Big\}\leq 2

\exp\left\{\frac{-\lambda^2}{2M^2 T}\right\}

\]

First express the event of interest in terms of the exponential martingale, then use the Kolmogorov-Doob inequality and after this choose the parameter \(\alpha\) to get the best bound.

# Exponential Martingale Bound

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