Past

There is a widespread use of mathematical tools in finance and its

importance has grown over the last two decades. In this talk we

concentrate on optimization problems in finance, in particular on

numerical aspects. In this talk, we put emphasis on the mathematical problems and aspects, whereas all the applications are connected to the pricing of derivatives and are the

outcome of a cooperation with an international finance institution.

As one example, we take an in-depth look at the problem of hedging

barrier options. We review approaches from the literature and illustrate

advantages and shortcomings. Then we rephrase the problem as an

optimization problem and point out that it leads to a semi-infinite

programming problem. We give numerical results and put them in relation

to known results from other approaches. As an extension, we consider the

robustness of this approach, since it is known that the optimality is

lost, if the market data change too much. To avoid this effect, one can

formulate a robust version of the hedging problem, again by the use of

semi-infinite programming. The numerical results presented illustrate

the robustness of this approach and its advantages.

As a further aspect, we address the calibration of models being used in

finance through optimization. This may lead to PDE-constrained

optimization problems and their solution through SQP-type or

interior-point methods. An important issue in this context are

preconditioning techniques, like preconditioning of KKT systems, a very

active research area. Another aspect is the preconditioning aspect

through the use of implicit volatilities. We also take a look at the

numerical effects of non-smooth data for certain models in derivative

pricing. Finally, we discuss how to speed up the optimization for

calibration problems by using reduced order models.

In 2008, Juhasz published the following result, which was proved using sutured Floer homology.

Let $K$ be a prime, alternating knot. Let $a$ be the leading coefficient of the Alexander polynomial of $K$. If $|a|

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I will introduce the theory of cluster categories after Amiot and Plamondon. For a quiver with a potential, the cluster category is defined as the quotient of the category of perfect dg-modules by the category of dg-modules with finite dimensional cohomologies. We can show the existence of the equivalence in the first talk as an application of the cluster category. I will also propose a definition of a counting invariant for each element in the cluster category.

The line graph operation, in which the edges of one graph are taken as the vertices of a new graph with adjacency preserved, is arguably the most interesting of graph transformations. In this survey, we will begin looking at characterisations of line graphs, focusing first on results related to our set of nine forbidden subgraphs. This will be followed by a discussion of some generalisations of line graphs, including our investigations into the Krausz dimension of a graph G, defined as the minimum, over all partitions of the edge-set of G into complete subgraphs, of the maximum number of subgraphs containing any vertex (the maximum in Krausz's characterisation of line graphs being 2).

The line graph operation, in which the edges of one graph are taken as the vertices of a new graph with adjacency preserved, is arguably the most interesting of graph transformations.  In this survey, we will begin looking at characterisations of line graphs, focusing first on results related to our set of nine forbidden subgraphs. This will be followed by a discussion of some generalisations of line graphs, including our investigations into the Krausz dimension of a graph G, defined as the minimum, over all partitions of the edge-set of G into complete subgraphs, of the maximum number of subgraphs containing any vertex (the maximum in Krausz's characterisation of line graphs being 2).

We develop a theory of optimal stopping problems under ambiguity in continuous time. Using results from (backward) stochastic calculus, we characterize the value function as the smallest (nonlinear) supermartingale dominating the payoff process. For Markovian models, we derive a Hamilton–Jacobi–Bellman equation involving a nonlinear drift term that describes the agent’s ambiguity aversion. We show how to use these general results for search problems and American Options.

Let $(Q',w')$ be a quiver with a potential given by successive mutations from a quiver with a potential $(Q,w)$. Then we have an equivalence of the derived categories of dg-modules over the Ginzburg dg-algebras satisfying the following condition: a simple module over the dg-algebra for $(Q',w')$ is either concentrated on degree 0 or concentrated on degree 1 as a dg-module over the

dg-algebra for $(Q,w)$. As an application of this equivalence, I will give a description of the space of stability conditions.

Abstract: Nonlinear models have been widely employed to characterize the

underlying structure in a time series. It has been shown that the

in-sample fit of nonlinear models is better than linear models, however,

the superiority of nonlinear models over linear models, from the

perspective of out-of-sample forecasting accuracy remains doubtful. We

compare forecast accuracy of nonlinear regime switching models against

classical linear models using different performance scores, such as root

mean square error (RMSE), mean absolute error (MAE), and the continuous

ranked probability score (CRPS). We propose and investigate the efficacy

of a class of simple nonparametric, nonlinear models that are based on

estimation of a few parameters, and can generate more accurate forecasts

when compared with the classical models. Also, given the importance of

gauging uncertainty in forecasts for proper risk assessment and well

informed decision making, we focus on generating and evaluating both point

and density forecasts.

Keywords: Nonlinear, Forecasting, Performance scores.

By means of a series of examples (Korteweg-de Vries equation, non-

linear stochastic heat equations and Navier-Stokes equation) we will show how it is possible to apply rough path ideas in the study of the Cauchy problem for PDEs with and without stochastic terms.

We study a generalized version of the signaling processoriginally introduced and studied by Argiento, Pemantle, Skyrms and Volkov(2009), which models how two interacting agents learn to signal each other andthus create a common language.

We show that the process asymptotically leads to the emergence of a graph ofconnections between signals and states which has the property that nosignal-state correspondance could be associated both to a synonym and aninformational bottleneck.

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After hep-th/0909.0483

I have reconstructed multiple palaeoecological records from sites across the British Isles; this work has resulted in detailed time series that demonstrate changes in vegetation, herbivore density, nitrogen cycling, fire levels and air temperature across an 8,000 year time span covering the end of the last glacial period. The aim of my research is to use statistics to infer the relationships between vegetation changes and changes in the abiotic and biotic environment in which they occurred. This aim is achieved by using a model-fitting and model-selection method whereby sets of ordinary differential equations (ODE) are ‘fitted’ to the time series data via maximum likelihood estimation in order to find the model(s) that provide the closest match to the data. Many of the differential equation models that I have used in this study are well established in the theoretical ecology literature (i.e. plant – resource dynamics and plant – herbivore dynamics); however, there are no existing ODE models of fire or temperature dynamics that were appropriate for my data. For this workshop, I will present the palaeoecological data that I collected along with the models that I have chosen to work with (including my first attempt at models for fire and temperature dynamics) and I hope to get your feedback on these models and suggestions for other useful modelling methods that could be used to represent these dynamics.