Sensitivity analysis of utility-based prices and risk-tolerance wealth processes

In the general framework of a semimartingale financial model and a utility function $U$ defined on the positive real line, we compute the first-order expansion of marginal utility-based prices with respect to a ``small'' number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases: \begin{tabular}@p97mm@ for any utility function $U$, if and only if the set of state price densities has a greatest element from the point of view of second-order stochastic dominance;for any financial model, if and only if $U$ is a power utility function ($U$ is an exponential utility function if it is defined on the whole real line). \end{tabular