Asymptotically optimal discretization of hedging strategies with jumps

In this work, we consider the hedging error due to discrete trading in models with jumps. Extending an approach developed by Fukasawa [In Stochastic Analysis with Financial Applications (2011) 331-346 Birkh\"{a}user/Springer Basel AG] for continuous processes, we propose a framework enabling us to (asymptotically) optimize the discretization times. More precisely, a discretization rule is said to be optimal if for a given cost function, no strategy has (asymptotically, for large cost) a lower mean square discretization error for a smaller cost. We focus on discretization rules based on hitting times and give explicit expressions for the optimal rules within this class.Comment: Published in at http://dx.doi.org/10.1214/13-AAP940 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic entropy and green speed for random walks on countable groups

We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91--112]), we give an alternative proof relying on a version of the so-called fundamental inequality (relating the rate of escape, the entropy and the logarithmic volume growth) extended to random walks with unbounded support.Comment: Published in at http://dx.doi.org/10.1214/07-AOP356 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Randomness on computable probability spaces - A dynamical point of view

We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff’s pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications

Optimal discretization of hedging strategies with directional views

We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectation-error criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived

An archaeological mystery revealed by radiocarbon dating of cross-flow nanofiltrated amino acids derived from bone collagen, silk, and hair: case study of the bishops Baldwin I and Radbot II from Noyon-Tournai

Excavations in the cathedral of Tournai revealed two sepultures, which were identified by the excavators as those of bishops because of their special location in the cathedral. One burial was assigned to Baldwin I, who died in AD 1068, because (1) a ring with the inscription "BAL" was found and (2) a funeral stone with text was present on top of the grave mentioning the name Baldewinus. The second burial probably belongs to Radbot II, who was the successor of Baldwin I, and died in AD 1098. Both burials contained textiles (silk), the skeleton, a wooden pastoral staff, and human hair was still present on the skull of what was presumed to be Radbot II. All the protein-containing materials were degraded and/or contaminated. Standard sample pretreatment methods were not able to remove all the contaminants. Single and double cross-flow nanofiltration of the hydrolyzed protein-containing materials were performed. The sample quality for radiocarbon dating was improved and C-14 data revealed interesting and surprising results. The C-14 dates of the wooden pastoral staff and permeate femur confirm that the skeleton and tomb belong to bishop Baldwin I. The C-14 dates of hair and permeate skull indicate that the skeleton may indeed belong to bishop Radbot II. The younger C-14 dates of the wooden pastoral staff and silk samples indicate a postburial disturbance of the site burial during the 12th-13th century

Standard Monomial Theory for Bott-Samelson Varieties of GL(n)

We construct an explicit basis for the coordinate ring of the Bott-Samelson variety Z_i associated to G = GL(n) and an arbitrary sequence of simple reflections i. Our basis is parametrized by certain standard tableaux and generalizes the Standard Monomial basis for Schubert varieties. Our standard tableaux have a natural crystal graph structure.Comment: Northeastern University, [email protected] AMSTeX amspp