14:15
14:15
14:15
Azema-Yor processes: three characterisation theorems
Abstract
We study the class of Azema-Yor processes which are of the form F(M_t)-f(M_t)(X_t-M_t), where F'=f, X_t is a semimartingale with no positive jumps and M_t is its running maximum. We show that these processes arise as unique strong solutions to the Bachelier SDE which we also show is equivalent to the DrawDown SDE. The proofs are greatly simplified thanks to (algebraic) group property of the set of AY processes indexed by functions. We then restrict our attention to the case when X is a martingale. It turns out that the AY martingales are the only local martingales of the form H(X_t,M_t) for a Borel function H. Furthermore, they can also be characterised by their optimal
properties: all uniformly integrable martingales whose maximum dominates a given target are dominated by an AY martingale in the concave ordering of terminal values. We mention how these results find direct applications in portfolio optimisation/insurance theory.
Joint work with Laurent Cararro and Nicole El Karoui
15:45
Pinning-depinning transition in Random Polymers
Abstract
Random polymers are used to model various physical ( Ising inter- faces, wetting, etc.) and biological ( DNA denaturation, etc.) phenomena They are modeled as a one dimensional random walk (Xn), with excursion length distribution
P(E1 = n) = (n)=nc, c > 1, and (n) a slowly varying function. The polymer gets a random reward, whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (Vn). Depending on the relation be- tween the mean value of the disorder Vn and the temperature, the polymer might prefer to stick on the interface (pinning) or undergo a long excursion away from it (depinning).
In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the 'Pinning' model and also its relation to other directed polymer models
15:45
Gradient estimate for the heat semi-group and heat estimates on H-type groups
Abstract
In this talk, we give the asymptotics estimates for the heat kernel and its gradient estimates on H-type groups. Moreover, we get gradient estimates for the heat semi-group.
14:15
Preferences and implicit risk measures
Abstract
We discuss some connections between various notions of rationality in the face of uncertainty and the theory of convex risk measures, both in a static and a dynamic setting.
15:45
A new combinatorial method for calculating the moments of Lévy area
Abstract
15:45
14:15
Existence of unique solutions for SDEs for individual driving paths.
Abstract
14:15
Risk Horizon and Rebalancing Horizon
Abstract
We analyze the impact of portfolio rebalancing frequency on the measurement of risk
over a moderately long horizon. This problem arises from an incremental capital charge recently
proposed by the Basel Committee on Banking Supervision. The new risk measure calculates
VaR over a one-year horizon at a high confidence level and assigns different
rebalancing frequencies to different types of assets to capture potential illiquidity.
We analyze the difference between discretely and continuously rebalanced portfolios in a simple model of asset dynamics by examining the limit as the rebalancing frequency increases. This leads to alternative approximations at moderate and extreme loss levels. We also show how to incorporate multiple scales of rebalancing frequency in the analysis
15:45
Lyapunov exponents of products of non-identically distributed independent matrices
Abstract
It is well known that the description of the asymptotic behaviour of products of i.i.d random matrices can be derived from the properties of the Lyapunov exponents of these matrices. So far, the fact that the matrices in question are IDENTICALLY distributed, had been crucial for the existing theories. The goal of this work is to explain how and under what conditions one might be able to control products of NON-IDENTICALLY distributed matrices.