Doob's Martingale Inequality - Application: Brownian Motion

Application: Brownian Motion

Let B denote canonical one-dimensional Brownian motion. Then

The proof is just as follows: since the exponential function is monotonically increasing, for any non-negative λ,

By Doob's inequality, and since the exponential of Brownian motion is a positive submartingale,

\begin{align} & \mathbf{P} \left \\ & = \mathbf{P} \left \\ & \leq \frac{\mathbf{E} \big}{\exp (\lambda C)} \\ & = \exp \left( \frac{\lambda^{2} T}{2} - \lambda C \right) \mbox{ since } \mathbf{E} \big = \exp \left( \frac{\lambda^{2} t}{2} \right). \end{align}

Since the left-hand side does not depend on λ, choose λ to minimize the right-hand side: λ = C / T gives the desired inequality.

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