# Probability, Stochastic Modelling and Financial Mathematics

## Group Members

These web pages describe the Probability, Stochastic Modelling and Financial Mathematics research group at the University of Leeds.

## Archive

Current news items can be found on the main page.

3 December 2015 — seminar talk
John Armstrong (King's College London)
Coordinate Free Stochastic Geometry with Jets.

A differential geometry perspective on the basic results of stochastic calculus (Ito's lemma, Fokker-Planck equation, Ito vs Stratonovich calculus). We answer the basic question "What does an SDE look like?" and use our pictures to understand the geometry underpinning stochastic differential equations.

26 November 2015 — seminar talk
Igor Evstigneev (University of Manchester)
Evolutionary Behavioural Finance.

The talk reviews a new research field that develops evolutionary and behavioural approaches for the modeling of financial markets. The main objective is to create a plausible alternative to the conventional Walrasian equilibrium theory based on the hypothesis of full rationality of market players. Rather than maximizing typically unobservable individual utility functions, traders/investors are permitted to have a whole variety of patterns of strategic behaviour depending on their individual psychology. The models considered in this field combine elements of evolutionary game theory (solution concepts) and stochastic dynamic games (strategic frameworks).

19 November 2015 — seminar talk
Giorgio Ferrari (Bielefeld University)
Continuous-Time Public Good Contribution under Uncertainty: A Stochastic Control Approach.

In this talk we consider continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner's optimal policy, we characterize it by necessary and sufficient stochastic Kuhn-Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation. We finally also provide a detailed analysis of the so-called free rider effect.

The talk is based on a joint paper with Frank Riedel and Jan-Henrik Steg.

12 November 2015 — seminar talk
Denis Denisov (University of Manchester)
Exit Times for Integrated Random Walks.

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time n. To show these asymptotics we develop a discrete potential theory for the integrated random walk. The talk is abased on a joint work with V. Wachtel
http://arxiv.org/abs/1207.2270

29 October 2015 — seminar talk
Jeremy Oakley (University of Sheffield)
Bayesian Calibration for Computer Models Using Likelihood Emulation.

I will first give a short overview of the field of Uncertainty Quantification: a variety of problems related to uncertainty in mathematical or 'computer' models of physical systems. I will then present some recent work (in collaboration with Ben Youngman) on calibration: finding model inputs such that the model outputs fit physical observations. Our approach is motivated by a case study involving a natural history model for colorectal cancer patients. The model is stochastic and computationally expensive, which inhibits evaluation of the likelihood function. We use a history matching approach, where we first exclude regions of input space where we can easily identify poor fits. We then construct an emulator (a fast statistical approximation) of the likelihood, which is used within importance sampling to sample from the posterior distribution of the computer model inputs.

20 July 2015 — seminar talk
Yana Belopolskaya (St Petersburg)
Probabilistic interpretation of systems of nonlinear parabolic equations.

Systems of nonlinear parabolic equations arise as mathematical models of many interesting phenomena in physics, biology, financial mathematics and other fields. We discuss probabilistic approaches to construct various types of the Cauchy problem solution to systems of nonlinear parabolic equations. In all cases we follow a three step procedure. Namely 1) we reduce the original Cauchy problem to a certain stochastic problem, 2) we solve the arising stochastic problem, 3) we prove that step 2) allows to obtain a solution of the original Cauchy problem, As a byproduct we construct stochastic representations for classical, generalized and viscosity solutions of the Cauchy problem for a large class of nonlinear parabolic systems.

7 May 2015 — seminar talk
Marco Iglesias (University of Nottingham)
A Bayesian level-set approach for geometric inverse problems.

We discuss a computational Bayesian framework for the solution of PDE-constrained inverse problems where the unknown is a geometric feature of the underlying PDE forward model. Geometric inverse problems of this type arise, for example, in the characterisation of subsurface formations where interfaces between geologic structures must be inferred from subsurface flow measurements. Another example emerges from Electrical Impedance Tomography (EIT) where the geometry of unhealthy tissue must be inferred from measurements of voltages associated to current distributions of a given configuration of electrodes. In this talk we introduce a level-set approach to infer and quantify the uncertainty of unknown geometric features in the aforementioned inverse problems within an infinite dimensional Bayesian framework. By means of state-of-the-art MCMC methods we describe numerical experiments which explore the posterior distribution that arises from the Bayesian level-set formulation.

Joint work with Matt Dunlop (Warwick), Yulong Lu (Warwick) and Andrew Stuart (Warwick)

30 April 2015 — seminar talk
Aretha Teckentrup (University of Warwick)
Multilevel Monte Carlo methods in Uncertainty Quantification.

The parameters in mathematical models for physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. By modelling the input parameters as stochastic processes, it is possible to quantify the uncertainty in the model outputs.

In this talk, we employ the multilevel Monte Carlo (MLMC) method to compute expected values of the model outputs, and we will focus in particular on two aspects. Firstly, we show how changing the mathematical model on the coarse levels makes the use of MLMC methods feasible also for problems exhibiting very fine scale features which cannot be resolved on the coarse levels. Secondly, we show how the multilevel framework can be extended to case of Markov chain Monte Carlo simulations.

19 March 2015 — seminar talk
Shiwei Lan (University of Warwick)
Adaptive geometric Monte Carlo using Gaussian process emulation for computation intensive models.

Bayesian statistics heavily relies on Markov Chain Monte Carlo (MCMC) methods for inference and prediction. When the posterior distribution is intractable, sampling from it becomes nontrivial. Random Walk Metropolis (RWM) is a classic MCMC algorithm, but well known to be slow in mixing. Recently, geometric information, e.g. gradient of log density in Hamiltonian Monte Carlo, metric and connection in Riemannian HMC, have been introduced to guide the sampler to efficiently explore the parameter space thus suppress the random walk nature of RWM. However, those geometric quantities usually require intensive computation for large data or complicated models, which affects the applicability of these geometric Monte Carlo methods. In this paper, we investigate the emulation by Gaussian process as a cheaper alternative to exact calculation of the geometry. The key idea is to approximate geometric quantities using Gaussian process conditioned on a carefully chosen design (training) set, which also determines the quality of the emulator. Realizing the difficulty of choosing a good design set a prior, we propose to use experiment design method to refine an arbitrarily initialized design online without destroying the convergence of the resulting Markov chain. The simulated and real examples demonstrate the significant improvement of our emulative geometric Monte Carlo methods over their full versions in terms of sampling efficiency and error reducing speed. Future directions are also discussed in the end including emulator design in high dimension, local predictor and parallelization.

5 March 2015 — seminar talk
Zdzislaw Brzezniak (University of York)
Strong and Weak Solutions to Stochastic Landau-Lifshitz Equations.

I will speak about the existence of weak solutions (and the existence and uniqueness of strong solutions) to the stochastic Landau-Lifshitz equations for multi (and one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire.

The talk is based on a joint works with B. Goldys and T. Jegaraj.

19 February 2015 — seminar talk
Dmitry Korshunov (Lancaster University)
Harmonic Functions and Stationary Distributions for Asymptotically Homogeneous Transition Kernels on $Z^+$.

We discuss a method for constructing positive harmonic functions for a wide class of transition kernels on $Z^+$. Sufficient conditions will be also presented under which these functions have positive finite limits at infinity. Further, we show how these results on harmonic functions may be applied to asymptotically homogeneous Markov chains on $Z^p$ with asymptotically negative drift. More precisely, assuming that Markov chain satisfy Cramer's condition, we demonstrate the tail asymptotics of the stationary distribution. In particular, these results allow to estimate the tail asymptotics for the stationary measure of a stable diffusion with asymptotically negative drift. The talk is based on a joint work with Denis Denisov and Vitali Wachtel.

5 February 2015 — seminar talk
Tiziano De Angelis (University of Manchester)
On the optimal exercise boundaries of swing put options.

Swing options are financial instruments designed primarily to allow for flexibility on purchase, sale and delivery of commodities in energy markets. They have features of American-type options with multiple early exercise rights and may be mathematically described in terms of multiple optimal stopping problems.

In particular we consider a swing option with put payoff, $N$ exercise rights and finite maturity, when the underlying asset's dynamics is described by a geometric Brownian motion according to the Black \& Scholes model. We prove that the optimal exercise of each right (except the last) is characterised in terms of two free-boundaries which are continuous functions of time and uniquely solve a system of coupled integral equations of Volterra-type. Finally the option's price formula is provided as the sum of a European part and an early exercise premium depending on the optimal exercise boundaries.

29 January 2015 — seminar talk
Angelos Dassios (LSE)
Brownian and other excursions and Parisian options.

We will review recent developments regarding the pricing of Parisian options and associated results on excursions. Parisian options are variations of barrier options. Barrier options involve events were a certain price level is exceeded. Fro Parisian options, the level has to be exceeded for a fixed period of time. The seminar will present the mathematical tools involved as well as applications. Historically, most of those results were presented in the form of inverse Laplace transforms. We will focus on an intuitive idea that enables us to calculate numerous quantities explicitly without numerical Laplace transform inversion.

26 January 2015 — Past Earth Network
Today the EPSRC-funded Past Earth Network, lead by PI Jochen Voss and CI Alan Haywood has officially started. The network aims to bring together the statistics and palaeo-climate communities. The themes of the network will be quantification of uncertainty within climate data and climate models, methods for model-data comparison and finally to use palaeo-climate data to assess and improve the performance of climate forecasts. See http://PastEarth.net/ for more information.
12 January 2015 — new group member
We are very happy to announce that Graham Murphy has joined the school as a Senior Teaching Fellow in financial and actuarial mathematics.
26 November 2014 — seminar talk
Tim Sullivan (University of Warwick)
Brittleness and Robustness of Bayesian Inference.

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors and others) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint with with Houman Owhadi and Clint Scovel (Caltech).

6 November 2014 — seminar talk
Bogdan Grechuk (University of Leicester)
Inverse Portfolio Problem and Restoring Risk Preferences.

In general, a portfolio problem minimizes risk (or negative utility) of a portfolio of financial assets with respect to portfolio weights subject to a budget constraint. The inverse portfolio problem then arises when an investor assumes that his/her risk preferences have a numerical representation in the form of a certain class of functionals, e.g. in the form of expected utility, coherent risk measure or mean-deviation functional, and aims to identify such a functional, whose minimization results in a portfolio, e.g. a market index, that he/she is most satisfied with. In this work, the inverse portfolio problem is solved for deviation measures and coherent risk measures. It can be used to restore investor's risk preferences based on the observable information.

30 October 2014 — seminar talk
Erik van Doorn (University of Twente)
Representations for the decay parameter of a birth-death process.

We discuss representations for the decay parameter of a birth-death process under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

23 October 2014 — seminar talk
Michela Ottobre (Edinburgh)
Diffusion limit for Random Walk Metropolis algorithm out of stationarity.

The Random Walk Metropolis algorithm is a Monte Carlo-Markov Chain method which creates a Markov chain which is reversible with respect to a given target distribution with Lebesgue density on $R^N$; it can hence be used to approximately sample the target distribution. When $N$ is large a key question is to determine the computational complexity of the algorithm as a function of $N$. We will explain how this question can be addressed by making use of diffusion limits for Markov chains.

16 October 2014 — seminar talk
Wei Yang (University of Strathclyde)
Introduction to Mean Field Games and Applications.

This talk first gives an introduction to the mean field games (MFG) methodology and then presents some of our recent results.

Mean field games were introduced around 2006 in the papers by J.M. Lasry and P.L. Lions in France and by M. Huang, R.P. Malhamé and P. Cains in Canada. Since then, there has been a fast growing literature on its further theoretical developments, applications and numerical solutions, etc.

Roughly speaking, the MFG method deals with problems involving large number interacting agents. Since the complexity of their analysis can become immense, the MFG method suggests to study corresponding tractable limiting problems as the number of the agents tends to infinity. Under certain general conditions, solutions can be found to the limiting problems. Then one can use a solution to the limiting problem to approximate the one to the original problem, namely, a solution to the limiting problem can be an epsilon-equilibrium of the original finite number agents problem.

9 October 2014 — seminar talk
Elena Issoglio (University of Leeds)
Multidimensional SDEs with distributional drift.

We consider a multidimensional stochastic differential equation with a time-dependent drift given by a distribution in a suitable class of Sobolev spaces with negative order. We show existence and uniqueness of a virtual solution for this singular SDE making use of a multidimensional Zvonkin type transform. In particular, we consider an auxiliary PDE whose coefficients are distributions and which is closely related to the Kolmogorov equation relative to the SDE. We show that there exists a function solution to this PDE and by using this solution we transform the singular SDE to a new SDE. For the latter we can show existence and uniqueness of a weak solution. This is a joint work with F. Flandoli and F. Russo.

2 October 2014 — seminar talk
State Estimation from Imperfect Knowledge - a Data Assimilation Toolbox.

We often learn about the real world by observing it and then interpreting those observations in the context of preconceived ideas. This is an interpretation of data assimilation - the task of combining imperfect observations of an evolving system - with imperfect prior knowledge - to estimate the state. This process also requires models linking the state to the observations.

This seminar will summarise the leading data assimilation methods used in the geosciences (including in weather prediction). All of these methods have roots in Bayes' Theorem although severe approximations have to be made for the practicalities. One such approximation leads to the EnKF (the ensemble Kalman Filter) which approximates uncertainty with a population of possible states. The EnKF suffers from sampling errors due to the finite size population and this seminar will show results from some adaptive schemes that may address this issue.

1 October 2014 — new group member
We are very happy to announce that Elena Issoglio has joined the school as a lecturer in financial mathematics.
8 May 2014 — seminar talk
Oxana Manita (Moscow State University)
On nonlinear Kolmogorov type equations.

It is well-known that the distribution of a solution of a stochastic differential equation satisfies a weak parabolic equation (so called Kolmogorov-Fokker-Planck equation). Equations of the same type arise independently in statistical mechanics and physics, for example, such as Vlasov equations. However, relatively few results concerning general equations are known.

In this talk a Cauchy problem for a parabolic equation in the space of probability measures with a rather general dependence of the coefficients on the solution will be considered. An analytic approach to the study of the well-posedness of this problem for unbounded coefficients will be developed.

(Joint work with Stanislav Shaposhnikov)

8 May 2014 — seminar talk
Beatrice Acciaio (London School of Economics and Political Science)
Arbitrage of the First Kind and Filtration Enlargements in Semimartingale Financial Models .

I will discuss the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition, in a general semimartingale financial model, under initial and under progressive filtration enlargements. In both cases, I will provide a simple and general condition which is sufficient to ensure this stability for any fixed semimartingale model. Furthermore, I will give a characterization of the NA1 stability for all semimartingale models.

(This talk is based on a joint work with C. Fontana and K. Kardaras.)

1 May 2014 — seminar talk
Chris Sherlock (University of Lancaster)
Optimising the pseudo-marginal random walk Metropolis.

Pseudo-marginal MCMC algorithms provide a general recipe for circumventing the need for target density evaluation when calculating the Metropolis-Hastings acceptance probability. Remarkably, replacing the target density with an unbiased stochastic estimator thereof still leads to a Markov chain with the desired stationary distribution. We review the pseudo-marginal random walk Metropolis algorithm and examine its overall efficiency in terms of both speed of mixing and computational time. Under a frequently encountered regime we identify the optimal acceptance rate and variance of the stochastic estimator. We also provide guidance for more general regimes.

27 March 2014 — seminar talk
Inexact Approximations for Doubly and Triply Intractable Problems.

Markov random field models are used widely in computer science, statistical physics and spatial statistics and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to an intractable likelihood function. Several methods have been developed that permit exact, or close to exact, simulation from the posterior distribution. However, estimating the marginal likelihood and Bayes' factors for these models remains challenging in general. This talk will describe new methods for estimating Bayes' factors that use simulation to circumvent the evaluation of the intractable likelihood, and compare them to standard ABC methods.

20 March 2014 — seminar talk
Viv Kendon (University of Leeds)
Quantum walks and their applications.

Quantum versions of random walks were inspired by their classical counterparts in the search for new quantum algorithms. They now have a wide range of applications from modelling physical systems and transport properties to universal quantum computation. I will give a gentle introduction to how their properties differ from classical random walk processes, and describe some of the applications, including recent work from my own group.

13 March 2014 — seminar talk
Simon Cotter (University of Manchester)
A Bayesian Approach to Shape Registration.

With the advent of more advanced prenatal scanning technologies, there is a need for diagnostic tools for certain congenital conditions. This problem amounts to finding the distance in shape space between a noisily observed scan of a particular organ, be it brain or heart etc, and a library of shapes of organs from babies that had particular conditions.

First we will introduce a function space analogue of the Random Walk Metropolis Hastings algorithm, an Markov chain Monte Carlo (MCMC) algorithm which allows us to sample from complex probability distributions on function space. Through some illustrative numerics, we will show that the function space version of this method is far more efficient than the more commonly used version.

We then frame the problem as a Bayesian inverse problem on function space, where the functions of interest relate to the geodesic flow fields that deform one shape into the other. This is analogous to finding the velocity field in a Lagrangian data assimilation problem. Using regularity results regarding the forward problem, we identify minimal-regularity priors in order to make the inverse problem well-posed. We then present some numerics for simple 2D examples on closed curves, which show how the posterior distributions on function space can be sampled using function space MCMC methods.

13 February 2014 — seminar talk
Iain Murray (University of Edinburgh)
Flexible models for high-dimensional probability distributions.

We bring the empirical success of deep neural network models in regression and classification to high-dimensional density estimation. Using the product rule, density estimation in D-dimensions can be reduced to D regression tasks. We tie these tasks together to improve computational and statistical efficiency, obtaining state-of-the-art fits across a wide range of benchmark tasks. I'll give a Bayesian data analysis example application from cosmology.

Work with Benigno Uria and Hugo Larochelle.
http://arxiv.org/abs/1310.1757

28 November 2013 — seminar talk
Yi Zhang (University of Liverpool)
Continuous-time Markov decision processes with total reward criteria.

In this talk I plan to describe some methods for studying continuous-time Markov decision processes (controlled pure jump processes) with the expected total reward (cost) criteria. Some recent development on the related optimality results (existence of optimal policies etc) will be reported, too.

14 November 2013 — seminar talk
Natalia Markovich (Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia)
Modeling Clusters of Extreme Values in Time Series.

The study of clusters of extreme values of a time series (exceedances over a sufficiently high threshold) is of fundamental interest in many applied fields including climate research, insurance and telecommunications. Usually clus- tering of exceedances corresponds to consecutive large losses occurring in a short period of time. Our main result states that limit distributions of cluster and inter-cluster sizes for a stationary sequence under specific mixing condition have geometric forms. The cluster size implies the number of con- secutive exceedances of the time series over a threshold and the inter-cluster size the number of consecutive observations running under the threshold. In Ferro & Segers (2003) the inter-cluster size normalized by the tail function is proved to be exponentially distributed. A geometric model of the limiting cluster size distribution was presented in Robinson & Tawn (2000) without rigorous proof. The inter-cluster size is also geometric distributed if clusters of exceedances are independent (Santos & Fraga Alves 2012). Recent results of the author will be presented [8]. It is shown that geometric limit distri- butions of both cluster and inter-cluster sizes can be represented by a level of a sufficiently high quantile of the underlying process used as the threshold and a so-called extremal index. The latter serves as a dependence measure. Its reciprocal has a simple interpretation as the limiting mean cluster size (Leadbetter et al. 1983). The presented models are in a good agreement with cluster and inter-cluster distributions of ARMAX and moving maxima pro- cesses and with real telecommunication data concerning packet traffic rates of Skype and IPTV peer-to-peer video applications. The suggested geometric distributions give rise to further modeling of distributions of return intervals (the time intervals between clusters) and durations of clusters widely used in seismology and climatology. It is derived that the limit distribution tail of the duration of clusters that is defined as a sum of the random number of the weakly dependent regularly varying inter-arrival times with tail index 0 < α < 2 is bounded by the tail of stable distribution. The results can be applied in numerous fields.

14 November 2013 — seminar talk
John Moriarty (University of Manchester)
Free boundary analysis of an irreversible investment model with finite fuel and random initial time with applications to electricity trading.

We employ methods from probability theory to provide a mathematical model for the small-scale trading of electricity. In particular we derive the Real Option value for two-party contracts where one party (agent A) decides at time tau to accept a financial sum P from the other party (agent B) and commits to physically supplying a unit of electricity to B at a maturity time T of B's choosing. The purpose of the contract is to hedge B's exposure to real-time market prices of electricity in the presence of time uncertainty over B's demand.

We will give a mathematical formulation in terms of an irreversible singular stochastic control problem mixed with an optimal stopping problem in an enlarged filtration, study some examples and provide a characterisation of the value of the contract, and of the optimal choice of tau and optimal inventory level policy for A. These are obtained via the analysis of free-boundary problems naturally associated to the control and stopping problems.

31 October 2013 — seminar talk
Neofytos Rodosthenous (University of Leeds)
Optimal stopping and free-boundary problems arising from mathematical finance (part 3/3).

Optimal stopping problems are an important and well-developed class of stochastic control problems. These kind of problems appear in various different research areas in sciences, one of which is mathematical finance. A great deal of the derivatives traded in the financial markets all over the world is of the so-called American-type, where the holder can exercise them at any time up to maturity. The rational prices of such contracts are therefore given by the values of their associated optimal stopping problems, which are considered under some martingale measures for the underlying risky asset price processes.

These optimal stopping problems are solved through their corresponding free-boundary problems for differential operators (e.g. Stefan's ice-melting problem in mathematical physics) and are then proved to be the solutions of the initial optimal stopping problems, through some standard verification arguments from stochastic analysis.

In this series of talks, an overview of some standard results will be given and some specific optimal stopping problems arising from mathematical finance will be discussed.

24 October 2013 — seminar talk
Neofytos Rodosthenous (University of Leeds)
Optimal stopping and free-boundary problems arising from mathematical finance (part 2/3).

Optimal stopping problems are an important and well-developed class of stochastic control problems. These kind of problems appear in various different research areas in sciences, one of which is mathematical finance. A great deal of the derivatives traded in the financial markets all over the world is of the so-called American-type, where the holder can exercise them at any time up to maturity. The rational prices of such contracts are therefore given by the values of their associated optimal stopping problems, which are considered under some martingale measures for the underlying risky asset price processes.

These optimal stopping problems are solved through their corresponding free-boundary problems for differential operators (e.g. Stefan's ice-melting problem in mathematical physics) and are then proved to be the solutions of the initial optimal stopping problems, through some standard verification arguments from stochastic analysis.

In this series of talks, an overview of some standard results will be given and some specific optimal stopping problems arising from mathematical finance will be discussed.

17 October 2013 — seminar talk
Alexey Piunovskiy (University of Liverpool)
Fluid approximation to controlled Markov chains with local transitions.

Fluid scaling is widely used in queuing theory, inventory, population dynamics etc because it usually leads to relatively simple deterministic models which describe the underlying stochastic systems accurately enough. In this talk, the idea of this approach will be illustrated on simple examples: firstly, for a fixed control strategy, the trajectories of the stochastic process converge almost surely to the (deterministic) solution of an ordinary differential equation; secondly, the solution of the original Bellman equation converges to the Bellman function of the corresponding deterministic optimal control problem. We shall discuss the rates of convergence and estimate the error terms for a special case of absorbing model. It should be emphasized that the straightforward fluid approximation sometimes fails to work. A concrete example will show that the deterministic Bellman function can be finite and well defined, whereas in the underlying stochastic model the Bellman function equals infinity. To avoid such situations, we propose the refined fluid approximation and show that it works for a wider class of models. If time permits, we can also discuss applications of this theory to information transmission, mathematical epidemiology and inventory.

10 October 2013 — seminar talk
Neofytos Rodosthenous (University of Leeds)
Optimal stopping and free-boundary problems arising from mathematical finance (part 1/3).

Optimal stopping problems are an important and well-developed class of stochastic control problems. These kind of problems appear in various different research areas in sciences, one of which is mathematical finance. A great deal of the derivatives traded in the financial markets all over the world is of the so-called American-type, where the holder can exercise them at any time up to maturity. The rational prices of such contracts are therefore given by the values of their associated optimal stopping problems, which are considered under some martingale measures for the underlying risky asset price processes.

These optimal stopping problems are solved through their corresponding free-boundary problems for differential operators (e.g. Stefan's ice-melting problem in mathematical physics) and are then proved to be the solutions of the initial optimal stopping problems, through some standard verification arguments from stochastic analysis.

In this series of talks, an overview of some standard results will be given and some specific optimal stopping problems arising from mathematical finance will be discussed.

25 July 2013 — seminar talk
Prof Galina Zverkina (Moscow, MIIT (Moscow State University of Railway Engineering))
Dynkin's identity and convergence rates in some queueing models.

An elementary and rigorous justification of Dynkin's identity with a generalised infinitesimal operator is provided for some queueing systems with discontinuous intensities of arrivals and serving. The basis is a simple complete probability formula. This is applied to the analysis of ergodicity, in particular, to polynomial bounds of convergence rate to stationary distribution. The emphasis is that a general version of Dynkin's formula is working for models where a formal infinitesimal operator may not exist and that it may be applied to processes which are formally non-Markov. The work is joint with A.Veretennikov.

7 March 2013 — seminar talk
Roger Tribe (University of Warwick)
Two Pfaffian Point Processes.

Coalescing (or annihilating) particle systems on the real line are an example of Pfaffian point process (similar to determinantal point processes). Previously Pfaffian point processes arose in random matrices (for example the real eigenvalues of the Gaussian orthonormal or Ginibre ensembles). This talk describes these two examples and a link between them. Joint work with Oleg Zaboronski (Warwick).

28 February 2013 — seminar talk
Gabriel Lord (Heriot-Watt University)
Numerical computation of stochastic travelling waves.

We discuss the numerical simulation of stratonovich SPDEs and the computation of travelling waves in the presence of stochastic forcing. We introduce a numerical scheme based on the weak solution and discuss how position and speed of the wave may be estimated. Final we discuss a technique that freezes the wave in the domain.

21 February 2013 — seminar talk
Mike Evans (University of Leeds)
Statistical mechanics of trajectories in driven systems.
11 February 2013 — seminar talk
Alexey Kulik (Kiev)
Evolutionary approach to the study of spectral properties of L_p generators of solutions to SDE's with jumps.

For L_p convergence rates of a time homogeneous Markov process, sufficient conditions are given in terms of an exponential phi-coupling. This provides sufficient conditions for L_p convergence rates and related spectral and functional properties (spectral gap and Poincare inequality) in terms of appropriate combination of local mixing' and recurrence' conditions on the initial process, typical in the ergodic theory of Markov processes. The range of applications of the approach includes processes that are not time-reversible. In particular, sufficient conditions for the spectral gap property for the Levy driven Ornstein-Uhlenbeck process are established.

24 January 2013 — seminar talk
Continuous-time Importance Sampling for Multivariate Diffusions.

Inference for multivariate diffusion processes is challenging due to the intractability of the dynamics of the process. Most methods rely on high frequency imputation and discrete-time approximations of the continuous-time model, leading to biased inference. Recently, methods that are able to perform inference for univariate diffusions which avoid time-discretisation errors have been developed. However these approaches cannot be applied to general multivariate diffusions.

Here we present a novel, continuous-time Importance Sampling method that enables inference for general continuous-time Markov processes whilst avoiding time-discretisation errors. The method can be derived as a limiting case of a discrete-time sequential importance sampler, and uses ideas from random-weight particle filters, retrospective sampling and Rao-Blackwellisation.

Joint work with Gareth Roberts, Giorgos Sermaidis and Krys Latuszynski.

6 December 2012 — seminar talk
Sotirios Sabanis (Edinburgh)
SDDEs in Financial Mathematics.

Motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors, we explore option pricing techniques under a stochastic delay differential equation (SDDE) framework. Moreover, we show that the delay effect is not too sensitive to time lag changes. The desirable robustness of the delay effect is demonstrated on several important financial derivatives as well as on the value process of the underlying asset. Furthermore, under the SDDE framework, the pricing of arithmetic Asian options is considered by employing techniques from the theory of comonotonicity. Finally, convergence results of Euler numerical schemes for SDDEs are obtained and their link to the theory of SDEs with random coefficients is highlighted. Thus, we show that Monte Carlo techniques can be correctly implemented under very mild conditions so as to closely approximate option prices.

All the above results are presented with the aim to demonstrate that the SDDE framework serves as a rich alternative approach to modelling the evolution of asset prices.

This talk consists of three parts. Part I is joint work with X. Mao [2], Part II is joint work with N. Mc Williams [3] and Part III is joint work with I. Gyongy [4].

22 November 2012 — seminar talk
Apostolos Kourtis (University of East Anglia)
Parameter Uncertainty in Portfolio Selection.

Under the classical mean-variance framework of Markowitz, optimal portfolio weights are a function of the expected returns on the risky assets available and the inverse covariance matrix of these returns. Parameter uncertainty arises from the fact that these two quantities are unknown in practice. Estimating them in finite samples generates extra risk to the portfolio choice process, known as estimation risk, and may result in portfolios that perform poorly out-of-sample. I review the statistical methods that have been proposed in the literature to deal with this problem. Focusing on my work in the area, I present two approaches to deal with parameter uncertainty. The first proposes new shrinkage-based methods to estimate the inverse covariance matrix. The second advocates the reduction of the number of assets in a manner that increases risk-adjusted returns. Both of these methods are shown to result in significant gains in out-of-sample portfolio performance.

15 November 2012 — seminar talk
Harry Zheng (Imperial College London)
Weak Maximum Principle for Stochastic Optimal Control.

In this talk we give a simple review on stochastic control theory and its applications. We mainly focus on necessary and sufficient stochastic maximum principle which can be expressed as a system of forward backward stochastic differential equations and some condition on the Hamiltonian for the optimal control. We use Clarke's generalised gradients and normal cones to characterise a weak version of the stochastic maximum principle and illustrate its usage with some examples.

8 November 2012 — seminar talk
Georgios Aivaliotis (Leeds)
Mean-Variance Stochastic Control.

I will try to give an overview of Mean-Variance optimisation problems. I will focus on continuous time Mean-Variance stochastic control problems and a method to circumvent the inapplicability of the Dynamic Programming Principle in these problems. I will look at the resulting HJB equations and ways to deal with their degeneracy. I will discuss different types of cost functionals and, if time permits, the idea of time consistent strategies.

1 November 2012 — seminar talk
Mauro Mobilia (Leeds)
Polarization, commitment and persuasion in the three-party constrained voter model.

The importance of relating "micro-level" interactions with "macro-level" phenomena in modelling social dynamics is now well established [1]. In this context, it has recently been argued that the seek for "consensus" limited by some form of "incompatibility" is a basic mechanism to explain the dynamics of cultural change and diversity [2]. Here, we will consider the mathematically amenable three-party constrained voter model (3CVM), which can be regarded as a bounded-compromise generalization of the celebrated two-state voter model [2,3]. In this individual-based opinion dynamics model, a finite population is composed of "leftists" and "rightists" that interact with "centrists" on a complete graph: a leftist and centrist can either become leftists with rate (1+q)/2 or centrists with rate (1-q)/2 (and similarly for rightists and centrists), where q denotes a selective bias towards extremism (q>0) or centrism (q<0). This system admits three absorbing fixed points and a "polarization" line along which a frozen mixture of leftists and rightists coexist. In the realm of the Fokker-Planck equation, and using a mapping onto a population genetics model, the fixation probability of ending in every absorbing state and the mean times for these events are obtained analytically and checked against stochastic simulations. Here, we show how fluctuations alter the mean field predictions in the limit of weak bias and large population size [3]. In the second part of the talk, to understand how a committed minority can resist a persuasive majority, we will consider a variant of the 3CVM in which some centrists are committed individuals ("zealots") that never change their state, whereas leftists and rightists are more persuasive than centrists [4]. The competition between a commitment and persuasion is thus characterized by studying the influence of the persuasion strength and zealots' density on the mean consensus time which is computed analytically and by simulations. This presentation is based on the recent references [3] and [4].

References:
[1] T. C. Schelling, J. Math. Social 1, 143 (1971); M. Granovetter, Am. J. Sociol. 78, 1360 (1973).
[2] see, e.g., R. Axelrod, J. Conflict Resolution 41, 203 (1997); "The complexity of cooperation", (Princeton University Press, 1997); G. Deffuant et al., Adv. Complex Syst. 3, 87 (2000).
[3] M. Mobilia, EPL (Europhysics Letters) 95, 50002 (2011).
[4] M. Mobilia, arXiv e-print: arXiv:1207.6270

4 October 2012 — seminar talk
Kody Law (University of Warwick)
Accurate Filtering for the Navier-Stokes Equation.
3 May 2012 — seminar talk
Matina Rassias (UCL)
Stochastic Differential Delay Equations and Applications.

In recent years an increasing interest in modelling real-life problems attracts the investigation of stochastic differential delay equations (SDDEs).

The mathematical formulation of SDDEs incorporates not only the idea of stochasticity but also the dependence of the state variable on the past states of the system under consideration. Two of the major research questions in the area of SDDEs are linked with the existence and uniqueness of the solution of the pertinent SDDE and the qualitative behaviour of the solution, as well.

In this talk, motivated by the two afore-mentioned questions, we are going to present:

a) tests for a wide class of non-linear SDDEs to have non-explosion solutions and

b) some moment and almost sure asymptotic estimations in order to identify their qualitative behaviour.

Finally, we will discuss how the theoretical results could be applied and extended in real-life problems such as problems arising from the area of the population dynamics.

24 April 2012 — seminar talk
Stephen Connor (York)
State-dependent Foster-Lyapunov criteria.

Foster-Lyapunov drift criteria are a useful way of proving results about the speed of convergence of Markov chains. Most of these criteria involve examining the change in expected value of some function of the chain over one time step. However, in some situations it is more convenient to look at the change in expectation over a longer time period, which perhaps varies with the current state of the chain.

This talk will review some joint work with Gersende Fort (CNRSTELECOM ParisTech), looking at when such state-dependent drift conditions hold and (perhaps more interestingly) what can be inferred from them.

19 April 2012 — seminar talk
Jon Pitchford (York)
Can simple networks model complex ecological interactions?.

An explosion of pollinator diversity on Earth accompanied the evolution of flowering plants, but surprisingly little is known about the mathematical and ecological rules that permit plants and their pollinators to co-exist. Could these rules help us understand similar patterns found where different components of a system work together, generating reciprocal benefits, from sub-cellular to social and economic networks? Previous attempts to answer this question, using complicated mathematics and supercomputers, disguised simpler and ecologically-meaningful patterns and mechanisms. In the models we present, simply counting the number of partners a species interacts with is a reliable predictor of biodiversity. This shifts the emphasis away from performing endless computations built on mathematical idealisations, and towards a more careful look at the biological detail structuring the natural world.

9 February 2012 — seminar talk
Jochen Voss (Leeds)
Maximum A Posteriori Estimators.

We consider the problem of estimating a function u from noisy measurements y of a known, possibly nonlinear, function G of the unknown u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field. Under natural conditions we show that the maximum a posteriori (MAP) estimator is well-defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior. The theory is illustrated with examples from the theory of conditioned diffusions.

(joint work with Andrew Stuart, Kody Law and Masoumeh Dashti)

26 January 2012 — seminar talk
Robert S. McKay, FRS FInstP FIMA (University of Warwick)
Dobrushin metric.

Dobrushin proved exponential convergence to a unique stationary measure for "weakly dependent" stochastic processes on networks of units, with respect to a class of summably Lipschitz test functions. In the case of a network of Polish spaces of bounded diameter, the topology on the associated space of probability measures can be metrised, facilitating a tidy presentation of his result. In his honour, I call the metric "Dobrushin's metric". Furthermore, it allows a quantification of the concept of "emergence". Although it may be difficult to calculate in general, a few simple examples of its computation will be given.

8 December 2011 — seminar talk
Konstantinos Zygalakis (Oxford Centre for Collaborative Applied Mathematics)
Qualitative Behaviour of Numerical Methods for SDEs and Application to Homogenization.

In this talk we will focus on some analytical tools that can been used to explain the qualitative behaviour of numerical methods for stochastic differential equations (SDEs). These tools (backward error analysis, modified equations) are well developed in the case of numerical methods for ordinary differential equations (ODEs), but such a systematic theory is currently lacking in the case of SDEs. We will start by quickly recapping how to derive modified equations in the case of ODEs, and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order weak methods, such as the Euler-Maruyama and the Milstein method. We will then focus on a specific example relating with the homogenization of passive tracers in a cellular flow in the limit of vanishing molecular diffusion, and explain how the theory of modified equations can guide us towards choosing a suitable numerical method for such a problem. This method will then be used for calculating quantities of interest, such as the effective diffusivity and exit times. If there is some time, we will briefly discuss the use of modified equations as a tool for constructing higher order methods for SDEs.

1 December 2011 — seminar talk
Jordan Stoyanov (Newcastle University)
Moment Analysis of Distributions: Recent Developments.

Any distribution with finite all moments is either unique (M-determinate) or nonunique (M-indeterminate). This property turns to be important for stochastic models in many areas including in finance and risk modelling. Some recent developments will be presented along classical criteria. The results will be illustrated by examples. There will be not so well-known facts which are surprising and even shocking. Challenging open questions will be outlined.

10 November 2011 — seminar talk
Vassili Kolokoltsov (University of Warwick)
Nonlinear Markov processes and kinetic equations.

We present a systematic discussion of the link between nonlinear positivity preserving equations (nonlinear kinetic equations) and Markov models of interacting particles. Approximating evolution of interacting particles is described by the first order linear approximation (evolution of derivatives with respect to initial data) to a nonlinear evolution, and the limit of the normalized fluctuations is described by the second order linear approximation to a nonlinear evolution (evolution of second derivatives with respect to initial data).

27 October 2011 — seminar talk
Mikhail Zhitlukhin (University of Manchester)
A Bayesian sequential testing problem of three hypotheses for Brownian motion.

We consider a sequential testing problem of three hypotheses that the unknown drift of a Brownian motion takes one of three values. We show that this problem can be solved by a reduction to an optimal stopping problem for local times of the observable process. For the case of large periods of observation, we derive integral equations for the optimal stopping boundaries and study their limit behaviour. The work can be regarded as a further step in the sequential testing problem of two hypotheses for Brownian motion, solved more than 30 years ago.

20 October 2011 — seminar talk
Ostap Hryniv (University of Durham)
Some limiting properties of non-homogeneous random walks with non-integrable increments.

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non-existence of moments for first-passage times and last-exit times. We demonstrate the benefit of the generality of our results by applications to some non-classical models, such as random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes.

(joint work with I.M.MacPhee, M.V.Menshikov and A.R.Wade)

29 September 2011 — seminar talk
Daniel Gruhlke (University of Bonn, Germany)
A Multilevel method for MCMC.

We present a Multilevel Markov chain Monte Carlo scheme which approximates expectation values with respect to infinite dimensional target measures. In infinite dimensional settings, standard Markov chain Monte Carlo methods such as the Metropolis algorithm have to face the problem to balance the dimensionality of the approximation (which lead to a smaller discetization error) and the number of steps of MCMC--steps that can be performed in a given time (which usually decreases with the dimension). Using different levels of discretization, we can avoid these problems and improve the order of convergence.

5th May 2011 — seminar talk
Dr Dirk Zeindler (York)
Central limit theorem for multivariate associated class functions with randomization — The cycle numbers with respect to the weighted probability measure.

B.M. Hambly, P. Keevash, N. O'Connell, and D. Stark have proven a central limit theorem for the logarithm of the characteristic polynomial $Z_n(x)$ of a permutation matrix. Associated class functions have been introduced as an extension of $Z_n(x)$ during some moment calculations. We now present here the proof of a central limit theorem for multivariate associated class functions with some randomization.

22nd March 2011 — seminar talk
Prof Anatoly Zhigljavsky (Cardiff)
Probabilistic models in stochastic global optimisation
9th March 2011 — seminar talk
Prof Ilya Goldscheid (Queen Mary, University of London)
On Random Walks in Random Environment in 1D

It is well known that random walks in one-dimensional random environment can exhibit sub-diffusive behaviour due to the presence of traps. In this talk I shall

1. Explain one (of numerous possible) definitions of a trap,
2. Show that the passage times for distinct traps are asymptotically independent exponential random variables
3. Describe the parameters of these random variables which appear to form, asymptotically, a Poisson process.
The above allows us to prove weak quenched limit theorems in the sub-diffusive regime where the contribution of traps plays the dominating role.
25th February 2011 — seminar talk
Dr Alet Roux (University of York)
Pricing of American and Bermudan options under proportional transaction costs

I will discuss the pricing, hedging and optimal stopping of American/Bermudan options under proportional transaction costs. A procedure for computing the bid and ask prices of the option will be demonstrated, as well as procedures for constructing the optimal stopping times and hedging strategies for the buyer and seller. In the presence of transaction costs, the optimal stopping times can differ for the writer and the holder of such options, so that the optimal exercise time for the holder is not necessarily the worst one for the writer. This gives rise to a situation not unlike a Nash equilibrium.

10th February 2011 — seminar talk
Dr Jochen Voss (University of Leeds)
The stationary distribution of discretised SPDEs

In this talk I will give a gentle introduction to the problem of how discretising a Stochastic Partial Differential Equation (SPDE) affects the equation's stationary distribution. All are welcome (and you don't need previous knowledge about SPDEs to attend the talk).

7th December 2010 — seminar talk
Dr Alexei Piunovskiy (University of Liverpool)
Controlled stochastic jump processes
18th November 2010 — seminar talk
Innovative Statistical Techniques for Modelling Clinical Trials

A large clinical trial for testing a new drug usually involves a large number of patients and is carried out in different countries using multiple clinical centres. A design of multicenter clinical trials consists of several stages including statistical design, predicting patient recruitment, randomization and drug supply chain processes.

The talk is devoted to the discussion of the advanced statistical techniques for modelling and predicting the behaviour of trial at different stages. The innovative analytic statistical methodology for predictive modelling of patient's recruitment is developed [1, 2, 3]. Patient's flows are modelled by using Poisson processes with random delays and gamma distributed rates. It allows predicting in time with confidence bounds the number of patients recruited in centres/regions using current recruitment data and also evaluating the optimal number of clinical centres needed to complete the trial before deadline with a given probability. The technique for predicting the number of patients randomized to different treatments and analyzing the impact of randomization process on the statistical power and sample size of the trial is also developed [4]. Basing on these results, an innovative risk-based statistical approach to modelling drug supply chain process is developed [3].

The software tools for patient's recruitment and drug supply modelling based on these techniques are developed. These tools are on the way of implementation and already led to substantial improvements of drug development process in GlaxoSmithKline.

References
[1] Anisimov, V.V., Fedorov, V.V., Modeling, prediction and adaptive adjustment of recruitment in multicentre trials. Statistics in Medicine, Vol. 26, No. 27, 2007, 4958-4975.
[2] Anisimov, V., Recruitment modeling and predicting in clinical trials, Pharmaceutical Out-sourcing. Vol. 10, Issue 1, March/April 2009, 44-48.
[3] Anisimov, V.V., Predictive modelling of recruitment and drug supply in multicenter clinical trials. Proc. of the Joint Statistical Meeting, Washington, USA, August, 2009, 1248-1259.
[4] Anisimov, V.V., Effects of unstratified and centre-stratified randomization in multicentre clinical trials. Pharmaceutical Statistics, 2010 (early view).
[5] Anisimov, V.V., Impact of stratified randomization in clinical trials, In: Giovagnoli A., Atkinson AC., Torsney B. (Eds), MODA 9 - Advances in Model-Oriented Design and Analysis. Physica-Verlag/Springer, Berlin, 2010, 1-8.
[6] Anisimov, V. Drug supply modeling in clinical trials (statistical methodology), Pharmaceutical Outsourcing, May/June, 2010, 50-55.

May 6, 2010 — seminar talk
Dr Georgios Aivaliotis (Leeds)
Mean-Variance Stochastic Control problems: A review
April 22, 2010 — seminar talk
Professor Mikhail B. Malioutov (Northeastern University, Boston)
Recovery of sparse active inputs of a system: a review
March 11, 2010 — seminar talk
Dr Krzysztof Latuszynski (Warwick)
Making black boxes out of black boxes - the Bernoulli Factory problem and its applications
February 26, 2010 — seminar talk
Dr Mauro Mobilia (Leeds)
Fixation in Evolutionary Games under Non-Vanishing Selection
February 12, 2010 — seminar talk
Professor Igor V. Evstigneev (Manchester)
Von Neumann-Gale Dynamical Systems and their Applications in Finance