04 - Local continuity - Tommi SOTTINEN

We propose the concept of Local Continuity that is somewhat related to directional continuity. DEFINITION: Let X and Y be, say, metric spaces. A function f from X to Y is locally continuous at point x in X if one can find an open set U(x) such that (i) x belongs to the closure of U(x), (ii) if x(n) converges to x in U(x) then f(x(n)) converges to f(x) in Y. The set U(x), the local continuity set of f at x, that tells the direction of continuity. If U(x) can be chosen to contain x then f is continuous at x. The concept was conceived during our study [Bender, C., Sottinen, T., and Valkeila, E. (2007): Pricing by hedging and no-arbitrage beyond semimartingales (under revision for Finance and Stochastics)] where we considered non-semimartingale pricing models that have non-trivial quadratic variation and a certain "small-ball property". It turned out that in these models one cannot do arbitrage with strategies that are continuous in terms of the spot and some other economic factors such as the running minimum and maximum of the stock. Unfortunately, this result does not extend to even simple strategies, when stopping times are involved. The reason is obvious: Stopping times are typically not continuous. However, local continuity turns out to be just what we need to prove our theorems, and the author is not aware of any reasonable stopping times that are not locally continuous. The talk is based on an ongoing joint work with C. Bender (Technical University of Braunschweig), D. Gasbarra (University of Helsinki), and E. Valkeila (Helsinki University of Technology). Tommi SOTTINEN. Reykjavik University. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1207750775297 (pdf) Ecouter l'intervention : Bande son disponible au format mp3 Durée : 49 mn