Mon, 09 Feb 2009
14:15
Oxford-Man Institute

Azema-Yor processes: three characterisation theorems

Dr Jan Obloj
(Oxford)
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

Mon, 09 Feb 2009
15:45
Oxford-Man Institute

Pinning-depinning transition in Random Polymers

Dr Nikolaos Zygouras
(Warwick)
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

Mon, 02 Feb 2009
14:15
Oxford-Man Institute

Preferences and implicit risk measures

Professor Hans Föllmer
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.

Mon, 19 Jan 2009
15:45
Oxford-Man Institute

A new combinatorial method for calculating the moments of Lévy area

Dr Daniel Levin
(Oxford)
Abstract
We present a new way to compute the moments of the Lévy area of a two-dimensional Brownian motion. This is a classical problem of great importance, originally solved by Lévy. Our approach uses iterated integrals and combinatorial arguments involving the shuffle product (joint paper with Mark Wildon, Swansea).

 

Mon, 19 Jan 2009
14:15
Oxford-Man Institute

Existence of unique solutions for SDEs for individual driving paths.

Professor Sandy Davie
(Edinburgh)
Abstract
Existence and uniqueness theorems for (vector) stochastic differential equations dx=a(t,x)dt+b(t,x)dW are usually formulated at the level of stochastic processes. If one asks for such a result for an individual driving Brownian path W then there is a difficulty of interpretation.

One solution to this is to use rough path theory, and in this context a uniqueness theorem can be proved (for a.e. W) for dx=b(x)dW if b has Holder continuous derivative. Another variant with a natural interpretation is dx=a(t,x)dt+dW where, if a is bounded Borel, uniqueness can be shown for a.e. W. The talk will explore the extent to which these two approaches can be combined.

Wed, 11 Mar 2009
14:15
Oxford-Man Institute

Risk Horizon and Rebalancing Horizon

Paul Glasserman
(Columbia)
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

Mon, 01 Dec 2008
15:45
Oxford-Man Institute

Lyapunov exponents of products of non-identically distributed independent matrices

Professor Ilya Goldschied
(London)
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.

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