DH 1st floor SR

Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots.

Many radar designs transmit trains of pulses to estimate the Doppler shift from moving targets, in order to distinguish them from the returns from stationary objects (clutter) at the same range. The design of these waveforms is a compromise, because when the radar's pulse repetition frequency (PRF) is high enough to sample the Doppler shift without excessive ambiguity, the range measurements often also become ambiguous. Low-PRF radars are designed to be unambiguous in range, but are highly ambiguous in Doppler. High-PRF radars are, conversely unambiguous in Doppler but highly ambiguous in range. Medium-PRF radars have a moderate degree of ambiguity (say five times) in both range and Doppler and give better overall performance.

The ambiguities mean that multiple PRFs must be used to resolve the ambiguities (using the principle of the Chinese Remainder Theorom). A more serious issue, however, is that each PRF is now 'blind' at certain ranges, where the received signal arrives at the same time as the next pulse is transmitted, and at certain Doppler shifts (target speeds), when the return is 'folded' in Doppler so that it is hidden under the much larger clutter signal.

A practical radar therefore transmits successive bursts of pulses at different PRFs to overcome the 'blindness' and to resolve the ambiguities. Analysing the performance, although quite complex if done in detail, is possible using modern computer models, but the inverse problems of synthesing waveforms with a given performance remains difficult. Even more difficult is the problem of gaining intuitive insights into the likely effect of altering the waveforms. Such insights would be extremely valuable for the design process.

This problem is well known within the radar industry, but it is hoped that by airing it to an audience with a wider range of skills, some new ways of looking at the problem might be found.

The first half of this seminar will discuss the hedging problem faced by a large sports betting agent who has to risk-manage an unwanted position in a bet on the simultaneous outcome of multiple football matches, by trading in moderately liquid simple bets on individual results. The resulting mathematical framework is that of a coupled system of multi-dimensional HJB equations.

This leads to the wider question of the numerical approximation of such problems. Dynamic programming with PDEs, while very accurate in low dimensions, becomes practically intractable as the dimensionality increases. Monte Carlo methods, while robust for computing linear expectations in high dimensions, are not per se well suited to dynamic programming. This leaves high-dimensional stochastic control problems to be considered computationally infeasible in general.

In the second half of the seminar, we will outline ongoing work in this area by sparse grid techniques and asymptotic expansions, the former exploiting smoothness of the value function, the latter a fast decay in the importance of principal components. We hope to instigate a discussion of other possible approaches including e.g. BSDEs.

The aim of this work is to show how to derive the electricity price from models for the

underlying construction of the bid-stack. We start with modelling the behaviour of power

generators and in particular the bids that they submit for power supply. By modelling

the distribution of the bids and the evolution of the underlying price drivers, that is

the fuels used for the generation of power, we can construct an spede which models the

evolution of the bids. By solving this SPDE and integrating it up we can construct a

bid-stack model which evolves in time. If we then specify an exogenous demand process

it is possible to recover a model for the electricity price itself.

In the case where there is just one fuel type being used there is an explicit formula for

the price. If the SDEs for the underlying bid prices are Ornstein-Uhlenbeck processes,

then the electricity price will be similar to this in that it will have a mean reverting

character. With this price we investigate the prices of spark spreads and swing options.

In the case of multiple fuel drivers we obtain a more complex expression for the price

as the inversion of the bid stack cannot be used to give an explicit formula. We derive a

general form for an SDE for the electricity price.

We also show that other variations lead to similar, though still not tractable expressions

for the price.

We consider a multidimensional structural credit model, where each company follows a jump-diffusion process and is connected with other companies via global factors. We assume that a company can default both expectedly, due to the diffusion part, and unexpectedly, due to the jump part, by a sudden fall in a company's value as a result of a global shock. To price CDOs efficiently, we use ideas, developed by Bush et al.

for diffusion processes, where the joint density of the portfolio is approximated by a limit of the empirical measure of asset values in the basket. We extend the method to jump-diffusion settings. In order to check if our model is flexible enough, we calibrate it to CDO spreads from pre-crisis and crisis periods.

For both data sets, our model fits the observed spreads well, and what is important, the estimated parameters have economically convincing values.

We also study the convergence of our method to basic Monte Carlo and conclude that for a CDO, that typically consists of 125 companies, the method gives close results to basic Monte Carlo."

In this talk I want to ask how to create a coherent mathematical framework for pricing and hedging which starts with the information available in the market and does not assume a given probabilistic setup. This calls for re-definition of notions of arbitrage and trading and, subsequently, for a ``probability-free first fundamental theorem of asset pricing". The new setup should also link with a classical approach if our uncertainty about the model vanishes and we are convinced a particular probabilistic structure holds. I explore some recent results but, predominantly, I present the resulting open questions and problems. It is an ``internal talk" which does not necessarily present one paper but rather wants to engage into a discussion. Ideas for the talk come in particular from joint works with Alex Cox and Mark Davis.

In this talk we will present results concerning the large scale long time behaviour of particles moving in a periodic (random) velocity field subject to molecular diffusion. The particle can be considered massless (passive tracer) or not (inertial particle). Under appropriate assumptions for the velocity field the large scale long time behavior of the particle is described by a Brownian motion with an effective diffusivity matrix K.

We then present some numerical algorithms concerning the calculation of the effective diffusivity in the limit of vanishing molecular diffusion (stochastic geometric integrators). Time permitting we will discuss the case where the driving noise is no longer white but colored and study the effects of this change to the effective diffusivity matrix.