 Solution

ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2011/2/a/solution

E [(M_{t}^{*})^{2}] \leq 4 E ∣ M_{t} ∣^{2} < 4 2^{- 2 n} = 2^{- 2 (n - 1)} .

Apply Markov's inequality to the nonnegative variable

(M_{t}^{*})^{2}

, at the threshold printed in the question:

P ((M_{t}^{*})^{2} \geq 2^{- (n - 1)}) \leq \frac{2 ^{- 2 (n - 1)}}{2 ^{- (n - 1)}} = 2^{- (n - 1)} .

The corresponding threshold for

M_{t}^{*}

itself is

2^{- (n - 1) /2}

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