Discrete-time expected-utility portfolio problem

For independent return innovations ${\xi }_n$ and a predictable process of holdings ${\theta }_n$, wealth obeysThe investor chooses the holdings to maximize expected utility of terminal wealth.

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 $-e^{-\gamma x-T{\mu }^2/(2{\sigma }^2)}$.

An income stream with per-period variance $b^2$ and correlation $\rho$ with a Gaussian traded price increment produces a hedge holding $-\rho b/\sigma$ in addition to speculative demand. The residual income variance is $b^2(1-{\rho }^2)$. With independent period pairs, the same conditional quadratic minimization applies at each date. A deterministic income fee changes the optimized value but not the hedge holding.

For terminal utility $U$ and independent return innovations,

If terminal utility is increasing and concave and each Bellman supremum is attained, backward induction shows that every remaining-horizon value function is increasing and concave. Concavity follows by combining optimal portfolios for two initial wealths with the same convex coefficient.