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Take the nondegenerate case $\sigma >0$ and an integer trading horizon $T$. For exponential utility, maximize the negative of the exponential loss, equivalently minimize its expectation. Conditional on past information, a predictable choice $\pi$ hasComplete the square:Thus the minimum conditional multiplier is $e^{-\kappa }$ with $\kappa ={\mu }^2/(2{\sigma }^2)$. Backward induction starts with terminal loss $e^{-\gamma x}$ and gives the minimal continuation loss $e^{-\gamma x-\kappa (T-t)}$. Independence of the next increment justifies the same conditional minimization for every history, not just deterministic trading plans. Hence exponential-utility trading with Gaussian increments givesThese are constant numbers of shares, not constant wealth fractions. Strategies with infinite exponential loss have expected utility $-\infty$ and cannot improve this finite value. If $\sigma =0$, the usual formula is inapplicable: with $\mu =0$ trading has no effect, while a nonzero deterministic increment admits unbounded riskless gains and no finite maximizing position.