= Exponential-utility trading with Gaussian increments
{title2=$\pi^*=\mu/(\gamma\sigma^2)$}
With a unit cash asset, independent <Gaussian> price increments of mean $\mu$ and positive <variance> $\sigma^2$, and exponential terminal utility $-e^{-\gamma x}$, the optimal <predictable> risky holding is the displayed constant number of shares. The one-step exponential-loss exponent is $-\gamma\mu\pi+\gamma^2\sigma^2\pi^2/2$; completing its square gives a minimal multiplier $e^{-\mu^2/(2\sigma^2)}$. <Backward induction> proves optimality among adapted strategies and gives value $-e^{-\gamma x-T\mu^2/(2\sigma^2)}$.
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