Oxford-Man Institute

 In the recent series of papers Kleban, Simmons, and Ziff gave a non-rigorous computation  (base on Conformal Field Theory) of probabilities of several connectivity events for critical percolation. In particular they showed that the probability that there is a percolation cluster connecting two points on the boundary and a point inside the domain can be factorized in therms of pairwise connection probabilities. We are going to use SLE techniques to rigorously compute probabilities of several connectivity events and prove the factorization formula.

Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they have also significant geometrical applications. Particular cases of planar maps are p-angulations, where each face (meaning each component of the complement of edges) has exactly p adjacent edges. Random planar maps have been used in theoretical physics, where they serve as models of random geometry.Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces.More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all p-angulations with n vertices. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. Both in the case p=3 and when p>3 is even, we prove that the resulting random metric spaces converge as n tends to infinity to a universal object called the Brownian map. This convergence holds in the sense of the Gromov-Hausdorff distance between compact metric spaces. In the particular case of triangulations (p=3), this solves an open problem stated by Oded Schramm in his 2006 ICM paper. As a key tool, we use bijections between planar maps and various classes of labeled trees

We consider nonlinear stochastic wave equations in dimension d\le 3.

Using Malliavin Calculus, we give upper bounds for the small eigenvalues of the inverse of two point densities.These provide a rate of degeneracy when points go close to each other.  Then, we analyze the consequences of this result on lower estimates for hitting probabilities. 

We define a notion of discrete Ricci curvature for a metric measure space by looking at whether "small balls are closer than their centers are". In a Riemannian manifolds this gives back usual Ricci curvature up to scaling. This definition is very easy to apply in a series of examples such as graphs (eg the discrete cube has positive curvature). We are able to generalize several Riemannian theorems in positive curvature, such as concentration of measure and the log-Sobolev inequality. This definition also allows to prove new theorems both in the Riemannian and discrete case: for example improved bounds on spectral gap of the Laplace-Beltrami operator, and fast convergence results for some Markov Chain Monte Carlo methods

We establish a large deviations principle for the block sizes of a uniformly random non-crossing partition. As an application we obtain a variational formula for the maximum of the support of a compactly supported probability measure in terms of its free cumulants, provided these are all non-negative. This is useful in free probability theory, where sometimes the R-transform is known but cannot be inverted explicitly to yield the density.

The vacant set is the set of vertices not visited by a random walk on a graph G before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit'

when the graph grows), and the relation of the problem to the random interlacement percolation.  I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.

Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies V (t) > t for all t > 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (S), where S := inf {t > 0 : Z_t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (S). We call this a dynamic Bessel bridge since S is not known at time 0 but is slowly revealed in time by observing Z. Our study is motivated by insider trading models with default risk. (this is a joint work with Luciano Campi and Albina Danilova)

Leveraged ETFs are funds that target a multiple of the daily return of a reference asset; eg UYG (Proshares) targets twice the daily return of XLF (Financial SPDR) and SKF targets minus twice the daily return of XLF.

 It is well known that these leveraged funds have exposure to realized volatility. In particular, the relation between the leveraged and the unleveraged funds over a given time-horizon (larger than 1 day) is uncertain and will depend on the realized volatility. This talk examines this phenomenon theoretically and empirically first, and then uses this to price options on leveraged ETFs in terms of the prices of options on the underlying ETF. The resulting model allows to model the volatility skews of the leveraged and unleveraged funds in relation to each other and therefore suggest an arbitrage relation that could prove useful for traders and risk-managers.