Past

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process.

The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process.

An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions, and rainfall. (Co-authors: D. C. Brody, Imperial College London, and A.

Macrina, King's College London and ETH Zurich. Downloadable at

www.mth.kcl.ac.uk.

Let E be an elliptic curve over the rationals. To get an asymptotic to the number of primes p

joint work with R Gurjar

Dirichlet to Neumann maps and their generalizations are exceptionally useful tools in the study of eigenvalue problems for ODEs and PDEs. They also have real physical significance through their occurrence in electrical impedance tomography, with applications to medical imagine, landmine detection and non-destructive testing. This talk will review some of the basic properties of Dirichlet to Neumann maps, some new abstract results which make it easier to use them for a wide variety of models, and some analytical/numerical results which depend on them, including detection and elimination of spectral pollution.

We call a family of permutations A in Sn 'intersecting' if any two permutations in A agree in at least one position. Deza and Frankl observed that an intersecting family of permutations has size at most (n-1)!; Cameron and Ku proved that equality is attained only by families of the form {σ in Sn: σ(i)=j} for i, j in [n].

We will sketch a proof of the following `stability' result: an intersecting family of permutations which has size at least (1-1/e + o(1))(n-1)! must be contained in {σ in Sn: σ(i)=j} for some i,j in [n]. This proves a conjecture of Cameron and Ku.

In order to tackle this we first use some representation theory and an eigenvalue argument to prove a conjecture of Leader concerning cross-intersecting families of permutations: if n >= 4 and A,B is a pair of cross-intersecting families in Sn, then |A||B|

The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).

The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) tries to maximise this quantity.

For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).