University of Edinburgh

An important and challenging problem in mathematical finance is how to choose a pricing measure in an incomplete market, i.e. how to find a probability measure under which expected payoffs are calculated and fair option prices are derived under some notion of optimality.

The notion of q-optimality is linked to the unique equivalent martingale measure (EMM) with minimal q-moment (if q > 1) or minimal relative entropy (if q=1). Hobson's (2004) approach to identifying the q-optimal measure (through a so-called fundamental equation) suggests a relaxation of an essential condition appearing in Delbaen & Schachermayer (1996). This condition states that for the case q=2, the Radon-Nikodym process, whose last element is the density of the candidate measure, is a uniformly integrable martingale with respect to any EMM with a bounded second moment. Hobson (2004) alleges that it suffices to show that the above is true only with respect to the candidate measure itself and extrapolates for the case q>1. Cerny & Kallsen (2008) however presented a counterexample (for q=2) which demonstrates that the above relaxation does not hold in general.

The speaker will present the general form of the q-optimal measure following the approach of Delbaen & Schachermayer (1994) and prove its existence under mild conditions. Moreover, in the light of the counterexample in Cerny & Kallsen (2008) concerning Hobson's (2004) approach, necessary and sufficient conditions will be presented in order to determine when a candidate measure is the q-optimal measure.

In this talk, a network topology is presented that allows both peer-to-peer and non-local traffic in a cellular based TDD-CDMA system known as opportunity driven multiple access (ODMA). The key to offering appropriate performance of peer-to-peer communication in such a system relies on the use of a routing algorithm which minimises interference. This talk will discuss the constraints and limitations on the capacity of such a system using a variety of routing techniques. A congestion based routing algorithm will be presented that attempts to minimize the overall power of the system as well as providing a measure of feasibility. This technique provides the lowest required transmit power in all circumstances, and the highest capacity in nearly all cases studied. All of the routing algorithms considered allocate TDD time slots on a first come first served basis according to a set of pre-defined rules. This fact is utilised to enable the development of a combined routing and resource allocation algorithm for TDD-CDMA relaying. A novel method of time slot allocation according to relaying requirements is then developed.

Two measures of assessing congestion are presented based on matrix norms. One is suitable for a current interior point solution, the other is more elegant but is not currently suitable for efficient minimisation and thus practical implementation.