Tuning an exponential random walk

MediumProbability Stochastic Processes~7m

Let $X_1, X_2, \dots$ be IID taking values $\pm 1$ with equal probability. Define $M_0 = 1$ and $M_n = M_{n-1}\,e^{X_n + c}$ for a constant $c$ . Find the unique $c$ that makes $\{M_n\}$ a martingale. (Give a decimal.)

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Martingale $\iff$ $\mathbb{E}[e^{X_n+c}]=1$ , i.e. $e^c\,\mathbb{E}[e^{X_n}]=1$ .
$\mathbb{E}[e^{X_n}]=\cosh 1$ , so $c=-\ln\cosh 1$ .

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-0.4338 (± 0.002)

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Asked at: Jane Street, Citadel

Tuning an exponential random walk

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