Pointwise regularity of the free boundary for the parabolic obstacle problem

We study the parabolic obstacle problem \lap u-u_t=f\chi_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1 and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that $f$ is Dini continuous, we prove that the set of regular points is locally a (parabolic) $C^1$-surface and that the set of singular points is locally contained in a union of (parabolic) $C^1$ manifolds

Mean-Dispersion Preferences and Constant Absolute Uncertainty Aversion

We axiomatize, in an Anscombe-Aumann framework, the class of preferences that admit a representation of the form V(f) = mu - rho(d), where mu is the mean utility of the act f with respect to a given probability, d is the vector of state-by-state utility deviations from the mean, and rho(d) is a measure of (aversion to) dispersion that corresponds to an uncertainty premium. The key feature of these mean-dispersion preferences is that they exhibit constant absolute uncertainty aversion. This class includes many well-known models of preferences from the literature on ambiguity. We show what properties of the dispersion function rho(dot) correspond to known models, to probabilistic sophistication, and to some new notions of uncertainty aversion.Ambiguity aversion, Translation invariance, Dispersion, Uncertainty, Probabilistic sophistication