Ricerc@Sapienza

We obtain a probabilistic proof of the local Lipschitz continuity for the optimal stopping boundary of a class of problems with state space [0; T] Rd, d 1. To the best of our knowledge this is the only existing proof that relies exclusively upon stochastic calculus, all the other proofs making use of PDE techniques and integral equations. Thanks to our approach we obtain ourresult for a class of diusions whose associated second order dierential operator is not necessarily uniformly elliptic. The latter condition is normally assumed in the related PDE literature.

It is known that the decision to purchase an annuity may be associated to an optimal stopping problem. However, little is known about optimal strategies, if the mortality force is a generic function of time and if the emph{subjective} life expectancy of the investor differs from the emph{objective} one adopted by insurance companies to price annuities. In this paper we address this problem considering an individual who invests in a fund and has the option to convert the fund's value into an annuity at any time.

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in R^{n+1} up to sets of null H^{n−1} measure. In particular, we prove (i) local finiteness of the (n−1)-dimensional Hausdorff measure of the free boundary, (ii) H^{n−1}-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at H^{n−1} almost every free boundary point.

We consider the thin-film equation with a prototypical contact-line condition modeling the effect of frictional forces at the contact line where liquid, solid, and air meet. We show that such condition, relating flux with contact angle, naturally emerges from applying a thermodynamic argument due to Weiqing Ren and Weinan E [Commun. Math. Sci. 9 (2011), 597–606] directly into the framework of lubrication approximation.