Approximate utility

Approximating arbitrage or a perfect hedge in a limit implies approximation in utility (formally, expected utilities are continuous in L 2 ), if and only if the utility function is bounded above and below by quadratics. We characterize when some special utility functions are continuous for all random consumptions with a common lower bound. Only linear, quadratic, and, in some cases, translated SAHARA utilities are continuous if there is no lower bound. We also characterize when a utility function defined on a convex subset can be extended to an L 2 -continuous nondecreasing and concave utility function on all of R . • Seemingly small errors can matter a lot to economic agents. • We characterize when small deviations in mean and variance do not matter a lot. • A small deviation does small damage if the overall payoff never gets small. • Applied to approximate diversification, hedging, portfolio choice, and arbitrage.