Two extensions of Thurston's spectral theorem for surface diffeomorphisms

Thurston obtained a classification of individual surface homeomorphisms via the dynamics of the corresponding mapping class elements on Teichm\"uller space. In this paper we present certain extended versions of this, first, to random products of homeomorphisms and second, to holomorphic self-maps of Teichm\"uller spaces.Comment: 11 page

A level-set approach for stochastic optimal control problems under controlled-loss constraints

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach. With this approach, the state constraints can be managed through an exact penalization technique

Time-dependent stabilization in AdS/CFT

We consider theories with time-dependent Hamiltonians which alternate between being bounded and unbounded from below. For appropriate frequencies dynamical stabilization can occur rendering the effective potential of the system stable. We first study a free field theory on a torus with a time-dependent mass term, finding that the stability regions are described in terms of the phase diagram of the Mathieu equation. Using number theory we have found a compactification scheme such as to avoid resonances for all momentum modes in the theory. We further consider the gravity dual of a conformal field theory on a sphere in three spacetime dimensions, deformed by a doubletrace operator. The gravity dual of the theory with a constant unbounded potential develops big crunch singularities; we study when such singularities can be cured by dynamical stabilization. We numerically solve the Einstein-scalar equations of motion in the case of a time-dependent doubletrace deformation and find that for sufficiently high frequencies the theory is dynamically stabilized and big crunches get screened by black hole horizons.Comment: LaTeX, 38 pages, 13 figures. V2: appendix C added, references added and typos correcte

Approximations of strongly continuous families of unbounded self-adjoint operators

The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.Comment: 22 pages, final version to appear in Commun. Math. Phy

A theoretical framework for the pricing of contingent claims in the presence of model uncertainty

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case.Comment: Published at http://dx.doi.org/10.1214/105051606000000169 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org