## MAXWELL INSTITUTE for MATHEMATICAL SCIENCES Probability Seminars Abstracts

### Venkat Anantharam: On the largest Lyapunov exponent for products of nonnegative random matrices

We derive an upper bound for the largest Lyapunov exponent of a Markovian random matrix product of nonnegative matrices. The bound is expressed as the maximum of a nonlinear concave function over a finite-dimensional convex set of probability distributions. The technique used is Markovian type counting. This is joint work with Reza Gharavi.

### Charles Bordenave: The radial spanning tree

In this talk, we present a new model of geometric random tree on the d- dimensional Euclidean space. This tree has a simple radial structure with the origin as root. This tree is motivated by applications in wireless mutlihop communications. When the set of points is a realisation of a homogeneous Poisson point process, some interesting functionals of this random tree are explicitly known, for example the distribution of the length of the edges or the mean degree of a node. We will also pay attention to the asymptotic behaviour of the tree, such as the path from a vertex to the origin, the shape of set of points of a given generation or also its topological ends. This work is a joint work with Francois Baccelli.

### Stanislav Volkov: 5x+1: how many go down?

I will talk about how probabilistic methods of analyzing randomly-labeled trees can provide an important insight on the 5x+1 analogue of the famous Collatz problem (3x+1), please see . Though no rigorous results about number theory will be proved in my talk, a number of properties of the trees with random labels will be rigorously established.

### Gareth Roberts: Perfect simulation of diffusions

The talk will describe joint work with Alexandros Beskos and Omiros Papaspiliopoulos on the simulation of diffusion processes without discretisation approximation. A number of examples to Monte Carlo calculations and likelihood based inference for diffusions will be briefly described.

### Denis Denisov: Asymptotics for first passage times of heavy-tailed Levy processes

We present exact asymptotics for the distribution of the first time a Levy process becomes negative, assuming heavy-tailed positive jumps and starting from positive level. We apply these results to find asymptotics for the distribution of the busy period in an M/GI/1 queue. This is joint work with Seva Sheer.

### Anton Shmatkov: The Rate of Convergence of Wong-Zakai Approximations for SDEs and SPDEs

We estimate the rate of convergence of the Wong-Zakai type of approximations for SDEs and SPDEs. Two cases are studied: SDEs in finite dimensional settings and evolution stochastic systems. The latter result is applied to the second order SPDEs of parabolic type and the filtering problem. Roughly, the result is the following. Let W_n be a sequence of continuous stochastic processes of finite variation on an interval [0,T]. Assume that for some α > 0 the processes W_n converge almost surely in the supremum norm in [0,T] to W with the rate n^{-κ} for each κ < α. Then the solutions u_n of the differential equations with W_n converge almost surely in the supremum norm in [0,T] to the solution u of the Stratonovich'' SDE with W with the same rate of convergence, n^{-κ} for each κ<α, in the case of SDEs and with the rate of convergence n^{-κ/2} for each κ<α, in the case of evolution systems and SPDEs. Finally, we verify that the two most common approximations of the Wiener process, smoothing and polygonal approximation, satisfy the assumptions made.

### Takis Konstantopoulos: Skorokhod reflection for real-valued functions on partially ordered sets

We study the so-called Skorokhod reflection problem (SRP) posed for real-valued functions defined on a partially ordered set (poset), when there are two boundaries, considered also to be functions of the poset. The problem is to constrain the function between the boundaries by adding and subtracting nonnegative nondecreasing (NN) functions in the most efficient way. We show existence and uniqueness of its solution by using only order theoretic arguments. The solution is also shown to obey a fixed point equation.
When the underlying poset is a σ-algebra of subsets of a set, our results yield a generalization of the classical Jordan-Hahn decomposition of a signed measure. We also study the problem on a poset that has the structure of a tree, where we identify additional structural properties of the solution, and on discrete posets, where we show that the fixed point equation uniquely characterizes the solution. Further interesting posets we consider are the poset of real n-vectors ordered by majorization, and the poset of n × n positive semidefinite real matrices ordered by pointwise ordering of the associated quadratic forms.
We say a function on a poset is of bounded variation if it can be written as the difference of two NN functions. The solution to the SRP when the upper and lower boundaries are the identically zero function corresponds to the most efficient or minimal such representation of a function of bounded variation. Minimal representations for several important functions of bounded variation on several of the posets mentioned above are determined in this paper. This is joint work with Venkat Anantharam.     Link to a preprint

### Anke Wiese: High order stochastic integrators for linear systems

We present high order numerical schemes for the strong solution of linear stochastic differential equations. These schemes are based on the Neumann and Magnus expansions. The Magnus integrator demonstrates an order of magnitude improvement in accuracy when compared to classical stochastic numerical integration schemes or the Neumann integrator. Moreover, in scenarios when the cost of computing the matrix exponential does not dominate, this improved accuracy is provided at the same computational cost. We also apply both the Neumann and Magnus integrators to a stochastic Riccati differential system that can be linearized. Both integrators demonstrate superior efficiency when compared to a non-linear stochastic Runge-Kutta method. (Joint work with Gabriel Lord and Simon Malham)

### Marc Lelarge: Tail asymptotics of subadditive processes and queueing networks

Tail asymptotics for the supremum of a random walk attracted a lot of interest in queueing theory or risk theory. In this talk, we show that these asymptotics can be extended to the case of the supremum of a subadditive process. This new result allows us to obtain large deviations results for queueing networks in their stationary regime. In the particular case of (max,plus)-linear recursions, the rate of exponential decay of the stationary solution is explicitly computed.

### Dima Korshunov: The key renewal theorem for a transient Markov chain

We consider a time-homogeneous Markov chain $X_n$ valued on real line. Suppose the chain is transient, that is, the chain generates a $\sigma$-finite renewal measure. We prove the key renewal theorem under condition that this chain has asymptotically homogeneous at infinity jumps and asymptotically positive drift.

### Lisa Müller: Stability classification of controlled queueing systems

We give an almost complete classification of ergodicity and transience conditions for a general multi-queue system with the following features: arrivals form Poisson streams and there are various routing schemes for allocating arrivals to queues; the servers can be configured in a variety of ways; completed jobs can feed back into the system; the exponential service times and feedback probabilities depend upon the configuration of the servers switching between service regimes is instantaneous. Several different levels of control of the service regimes are considered.
Our results for the $N$-queue system require randomisation of service configurations. This will be illustrated by showing the result in detail for two queues. We also provide some examples of queueing systems which are covered by our model.

### Paavo Salminen: Tanaka formula for symmetric Lévy processes

Starting from the potential theoretic definition of the local times of a Markov process -- when these exist -- we obtain a Tanaka formula for the local times of symmetric Lévy processes. The most interesting case is that of the symmetric α-stable Lévy process (for α ∈ (1,2]) which is studied in detail. In particular, we determine which powers of such a process are semimartingales. These results complete, in a sense, the works by K. Yamada, and Fitzsimmons and Getoor.

### Eugene Shargorodsky: Boundary value problems for elliptic pseudodifferential operators arising in the theory of Markov processes

I will use the theory of boundary value problems for elliptic pseudodifferential operators to discuss the well known difficulties which arise in the study of Markov processes with state spaces having a boundary and which are largely unresolved, especially in the case of general (Wentzell) boundary conditions. The talk will be aimed at probabilists with possibly little or even no prior knowledge of the theory of pseudodifferential operators.

### Pavel Grigoriev: Risk measures on atomless probability spaces

In the first half of the talk I shall survey the basic facts about coherent and convex risk measures (in the contexts of the talk the risk measures are a class of some non-linear functionals on the space of uniformly bounded random variables).
Second part of the talk will be devoted to the recent result of A.Cherny and P.Grigoriev, who proved that on an atomless probability space a risk measure is law invariant iff it is dilatation monotone (i.e. monotone with respect to the conditional expectations over all sigma-subalgebras). In some sense this result could be viewed as an inverse theorem to the characterization of the law-invariant risk measures due to Kusuoka and Follmer and Schied.
No preliminary knowledge (but basic probability and analysis) is assumed.

### Maury Bramson: Exclusion processes in one and higher dimensions

Exclusion processes form one of the major classes of interacting particle systems. There, particles on a lattice execute independent random walks in continuous time, except when the target site is already occupied, in which case the particle remains at the original site.
Many results exist for the lattice Z. In particular, the equilibria of such exclusion processes are in many cases well understood. Little is presently known, however, for Z^d, d>1 . We will review the behavior of the exclusion process on Z and with the remaining time present the foundation of a theory for d>1 .

### Alexandre Proutière: Mean field analysis of random multi-access algorithms

In this talk, using mean field techniques, we present a performance analysis of random multi-access algorithms, such as the exponential back-off algorithm, in the case of a finite number of saturated or non-saturated sources. We prove a decoupling property when the number of sources grows large, and present some applications of this property.

### Xiaoli Chen: Solutions of stochastic differential inclusions

Stochastic differential inclusions(SDIs) represent an important generalization of the notion of stochastic differential equations(SDEs). In this talk we will discuss SDIs on domain of $\bR^d$. Some new results on solvability and approximations of SDIs will be presented. The existence of solutions is proved by monotonicity method in terms of minimizing certain convex functionals and in this way solutions are approximated by implicit schemes. In this presentation, a result on the rate of convergence will be discussed.

### Seva Shneer: Random multiple-access protocols with spatial interactions: discrete and continuous cases.

We adopt the classical ALOHA scheme to the case when the users in a multiple-access system have local interactions given by some neighbourhood relations. We prove that the fluid limits of the system's workload satisfy a certain differential equation and, via an analysis of this differential equation, obtain conditions sufficient for the system to be stable. We also discuss the behaviour of the fluid limits. This is a joint work with Charles Bordenave and Serguei Foss.

### Serguei Foss: On lower bounds and equivalences for distributional tails of randomly stopped sums

I present a short new proof of the following result.
{\bf Theorem.} Let $F$ be a distribution on the positive half-line and $\overline{F}(x) = F((x,\infty ))$. Assume that $F$ is {\it heavy-tailed}, that is: for any $c>0$, $$\int_0^{\infty} e^{cx} F(dx) = \infty.$$ Let $\xi_1,\xi_2,\ldots$ be i.i.d. sequence of random variables with a common distribution $F$. Let $\tau$ be an independent {\it light-tailed} counting random variable and $F^{*\tau}$ a distribution of $S_{\tau} = \sum_1^{\tau} \xi_i$.
Then $$\liminf_{x\to\infty} \frac{\overline{F^{*\tau}}(x)}{\overline{F}(x)} = {\bf E} \tau.$$
\vspace{0.5cm}
I also provide various comments on this limit and on relative problems.

### Moez Draief: Threshold for Virus Spread on Networks

We consider epidemics on finite undirected graphs and study the relationship between the epidemics outcome, the infection parameters, and properties of the graph. We establish thresholds on certain classes of graphs with homogeneous node degrees. In addition, we obtain thresholds for epidemics on power-law graphs.

### Alexander Veretennikov: On Doeblin-Doob and other mixing conditions for Markov processes

In his seminal Stochastic Processes (1953), J.L.Doob, in particular, gave a detailed classification of various conditions for a homogeneous Markov process to have an (exponential) convergence to equilibrium distribution. In the literature, the most general condition used in this theory is called Doeblin-Doob's condition. Some of those conditions provide a uniform (exponential) convergence on some classes of processes. Other conditions allow only an (always exponential) convergence rate for a single process. Doob's most general case that gives a uniform convergence on some class is most widely used, unlike Doeblin-Doob's one. Recently it was understood that there is a natural more general condition that also provides a uniform convergence on some appropriate class of processes; moreover, this condition also gives a more natural bound for the exponent power in this rate. In the talk, this condition will be discussed with applications to mixing for approximation schemes of stochastic differential equations under a good recurrence.

### Peter Mörters: Intersections of Random Walks in Supercritical Dimensions

In high dimensions two independent simple random walks have only a finite number of intersections. I discuss recent results on the upper tail behaviour of the number of intersections, obtained in collaboration with Xia Chen (Knoxville)

### Vassili Kolokoltsov: Fractional Dynamics and Continuous Time Random Walks

Functional limit theorem for continuous-time random walks (CTRW) are found in the general case of dependent waiting times and jump sizes that are also position dependent. The limiting anomalous diffusion is described in terms of fractional dynamics that is often considered by physicists as a natural model for Hamiltonian chaos. Probabilistic interpretation of generalized fractional evolution is given in terms of the random time change (subordination) by means of hitting time processes.

### Thomas Liniger Multivariate Hawkes Processes and Applications

We consider multivariate, marked Hawkes processes and give the conditions for sub-critical behavior and a direct proof based on [Brémaud and Massoulié, 1996]. To achieve this we use Poisson embeddings, which can be used to derive further results concerning stability under initial conditions. Hawkes processes have been widely applied in earthquake modelling and have found more recent applications in finance, such as pricing of credit derivatives [Errais, Giesecke and Goldberg, 2007].

### Vladimir Rykov: Queueing Networks with Transparent Calls

In some applications of queueing networks the situation arises when all calls of the same type can be served simultaneously as if they were a single call. One of the main applications of this type of model is in queueing networks with multipoint connection protocols, which are used in TV broadcasting, video-on-demand, some types of teleconferencing, etc.. Analagous service principles are probably realised in neuron networks, where an impulse caused by some irritant is delivered to all executives for its organs simultaneously. traditional approaches to the analysis of complex telecommunications networks are based on modelling of stochastic networks (SN) and reversible Markov processes, which permit us to obtain the steady state probabilities in product form. The reversibility approach is not applicable directly for stochastic networks with general service time distributions, because of the absence of a theory of reversible processes on general state spaces. In this case the product form for the staedy state distribution is usually obtained by the phase method approach. This approach consists of approximation of any distribution by the mixture of convolutions of exponential distributions, and the fact that these mixtures are dense in the class of distributions on the positive reals. But the weak convergence arguments necessary for this are only proved for the case of a finite state space. An alternative approach to proving the product form for stochastic networks with general service time distributions was proposed in a number of papers by Rykov et al. This approach is based on direct investigation of Markov processes with additional variables which describe the stochastic network under consideration. Fuller Abstract

### Florin Avram:

The one dimensional exact and asymptotic exit/ruin theory have been developed by Cramer and Lundberg as an application of Laplace's method. In several dimensions this method breaks down; exact solutions are very rarely available, and even asymptotic expansions (beyond the large deviations logarithmic approximation) are not fully studied.
On the other hand, exact and asymptotic results are available for a certain class of "degenerate" multidimensional Levy processes which appear in proportional reinsurance problems, and which may be reduced to one-dimensional first-passage problems with linear barriers.
(Joint work with Zbigniew Palmowski and Martijn Pistorius.)

### Stephen Connor: Coupling: Co-adapted vs. Maximal

Coupling is an interesting probabilistic technique, with many applications. The existence of maximal couplings for general stochastic processes has been known for some time, but such couplings are often not co-adapted (and hence usually rather unintuitive). In this talk I will introduce the concepts of maximal and co-adapted couplings, before looking at how good co-adapted couplings can possibly be in the case of two simple stochastic processes: Euclidean Brownian motion and a simple symmetric random walk on the hypercube. This topic is interesting in its own right, but also has consequences for the possible efficiency of perfect simulation algorithms such as Coupling from the Past.

### Mike Steel: Combinatorial and stochastic approaches in phylogenetics

Phylogenetics is the reconstruction and analysis of 'evolutionary' trees and graphs in biology (and related areas of classification such as linguistics). Several areas of mathematics and statistics are playing an increasingly important part in the underlying theory. I will describe some of the ways in which concepts from combinatorics and probability have helped provide insights into certain fundamental questions relevant to evolutionary biology, and outline some open problems.

### Nikolai Leonenko: Multifractality of Products of Geometric Ornstein-Uhlenbeck Type Processes

This is joint work with V.V. Anh (Queensland University of Technology, Brisbane) and N.-R. Shieh (National Taiwan University, Taipei). We consider multifractal products of stochastic processes as de.ned in Man- nersalo et al. (2002), but we provide a new interpretation of the conditions on the mean, variance and covariance functions of the resulting cumulative processes in terms of the moment generating functions. We show that the logarithms of the corresponding limiting processes have an in.nitely divisible distribution such as the gamma and variance gamma distributions (resulting in the log-gamma and log-variance gamma scenarios respectively), the inverse Gaussian and normal inverse Gaussian distributions (yielding the log-inverse Gaussian and log-normal inverse Gaussian scenarios respectively). We describe the behavior of their q-th order moments and Rï¿½nyi functions, which are non- linear, hence displaying their multifractality. A property on the dependence structure of the limiting processes, leading to their possible long-range depen- dence, is also obtained.
We consider very diï¿½erent scenarios such as the log-gamma and log-inverse Gaussian scenarios as typical examples of our approach. We should also note some related results by Barndorï¿½-Nielsen and Schmiegel (2004) who introduced some Lï¿½vy-based spatiotemporial models for parametric modelling of turbu- lence. Log-in.nitely divisible scenarios related to independently scattered ran- dom measures were introduced in Bacry and Muzy (2003) and others. We should note that Chris Heyde (1999) proposed to use a multifrcality into risky asset model with fractal activity time (see also Heyde and Leonenko (2005)).

### Nikita Vvedenskaya: Differential Equations Arising in Network Problems

The talk focuses on large queueing systems and networks with dependent queues. In particular, we are interested in the large-time performance and stationary distribution of such networks. These problems are often rather complicated, and one of the ways to study them is via an asymptopical approach.
In case where the network is guided by a Markov process and the number of its nodes $N$ is very large, the limiting situation, as $N\to infty$, may be described by differential equations. The solution to these equations gives an accurate assessment of the network performance. I present several examples where such aproach turnes to be successful.
First, we consider fast Jackson networks with dynamic routing. T he simplest system with dynamic routing was given by Dobrushin, Karpelevich and Vvedenskaya (and independently, Mitzenmacher): it contains $N$ identical servers with IID exponential service times and a Poisson flow of tasks. Upon its arrival, each task randomly selects $m$ servers and is directed to the one with a shorter queue. The limiting situation is described by a rather simple infinite system of ODEs, with a unique globally attracting fixed point. The probability of long queues in the limiting system decreases superexponentially. Numerical simulations show that in the system with finite $N$ the queue length distribution is close to the limit one. Later, Martin--Suhov and Suhov--Vvedenskaya considered open Jackson-type networks where each node contains $N$ identical servers and a task selects several servers (from one or several nodes) and is directed to the one with a shorter queue.
Finally, in a random multiple access system with $N$ users and ALOHA-like protocol, each user tries to transmit its message with a frequency determined by its back-off stage (the stage is changed after each transmission.) As $N\to \infty$, the system performance is described by a solution to a system of ODE with one or several fixed points. In case of several solutions the original system is treated as 'metastable'.

### Goran Peskir: The British Option

We present a new put/call option where the buyer may exercise at any time prior to maturity whereupon his payout is the best prediction' of the Euro- pean payout under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is the protection feature which is key to the British option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements), he can substitute the true drift with the contract drift and minimise his losses. With the contract drift properly selected the British put option becomes a more buyer friendly' alternative to the American put: when stock price move- ments are favourable, the buyer may exercise rationally to very comparable gains; when price movements are unfavourable he is afforded the unique pro- tection described above. Moreover, the British put option is always cheaper than the American put. In the final part we present a brief review of optimal prediction problems which preceded the development of the British option. This is a joint work with F. Samee (Manchester).

### Terry Lyons: Old work of Krylov and new uses for Ito's map

It is an interesting exercise to compute the iterated integrals of Brownian Motion and to calculate the expectations (of polynomial functions of these integrals).

Recent work on constructing discrete measures on path space, which give the same value as Wiener measure to certain of these expectations, has led to promising new numerical algorithms for solving 2nd order parabolic PDEs in moderate dimensions. Old work of Krylov associated finitely additive signed measures to certain constant coefficient PDEs of higher order. Recent work with Levin allows us to identify the relevant expectations of iterated integrals in this case, leaving many interesting open questions and possible numerical algorithms for solving high dimensional elliptic PDEs.

Abstract

### Juan Carlos Pardo Millan: Positive self-similar Markov processes

Positive self-similar Markov processes are positive real-valued strong Markov processes satisfying the scaling property with some index α > 0. These processes are to be found in many areas of probability theory. To give just a few examples, let us mention: the continuous state branching process which is associated to a self-similar Lévy tree and whose genealogy may be described by the beta-coalescent, the size of a tagged individual in the self-similar branching Markov chain, the so called tagged fragment of a self-similar fragmentation. These processes also appear as the limit of local structure of random quadrangulations. In this talk, we survey some recent results on this topic.

This is joint work with M.E. Caballero, L. Chaumont and V. Rivero.

### Christina Goldschmidt: Behaviour near the extinction time in stable fragmentation processes

The stable fragmentations are particularly nice examples of the self-similar fragmentation processes introduced by Bertoin. They are derived by looking at the masses of the subtrees formed by discarding the the parts of a stable continuum random tree below height t, for t ≥ 0. Such a fragmentation possesses a parameter α which lies between -1/2 and 0, called the index of self-similarity; heuristically, any block of mass m splits at a rate proportional to mα. Since α is negative, this means that smaller blocks split faster than larger ones. Indeed, small blocks split faster and faster until they are reduced to dust (blocks of size 0). The whole state is entirely reduced to dust in some almost surely finite time zeta, called the extinction time. We give a detailed limiting description of the distribution of a stable fragmentation of index α as it approaches its extinction time.

This is joint work with Benedicte Haas of Universite Paris-Dauphine.

### Ben Hambly: Local limit theorems for simple random walks on sequences of graphs

In recent years there have been invariance principles proved for random walk on supercritical percolation clusters and for random walks on fractal graphs. We show that these theorems can be refined to yield local limit theorems for the convergence of the transition densities of the random walks to those of the appropriate heat kernel on the limit of the seqence of graphs.

### Mokshay Madiman: Information inequalities for joint distributions, with interpretations and applications (tentative)

Upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies. These inequalities generalize Shannon's chain rule for entropy as well as inequalities of Han, Fujishige and Shearer. A duality between the upper and lower bounds for joint entropy is developed, and connections to structural properties of the entropy function are explored. The new inequalities are applied to obtain new results in combinatorics, such as bounds on the number of independent sets in an arbitrary graph and the number of zero-error sourcechannel codes, as well as new determinantal inequalities in matrix theory. A new inequality for relative entropies is also developed, along with interpretations in terms of hypothesis testing.

### Ari Arapostathis: Uniform stability of controlled Markov processes and applications

We consider controlled Markov Processes modeled by Itô stochastic differential equations in Rd, i.e., controlled diffusions. A control is called stable if the corresponding process is positive recurrent. It is well known that the invariant probability measures of a positive recurrent diffusion can be characterized via an embedded Markov chain which lives on the boundary of an annulus in Rd, a method due to R. Hasminskii. A controlled diffusion is called uniformly stable if the first return times to the annulus are uniformly integrable, with respect to all stationary Markov controls, when the initial distribution of the diffusion is taken as the invariant distribution of the embedded chain. This definition is due to Vivek Borkar.
• The first part of the talk will be a tutorial on ergodic properties of diffusions, including some recent results of Nikolai V. Krylov and V.I. Bogachev.
• Subsequently, we present some key new results:
(a) if each stationary Markov control is stable, then the controlled diffusion is uniformly stable, and
(b) that uniform boundedness of the mean hitting time of some bounded domain is sufficient for uniform stability.
• Next, we present equivalent characterizations of uniform stability, either in terms of the collection of invariant probability measures, or via stochastic Lyapunov functions.
• Lastly, we discuss applications in optimal control.

### Michael Zazanis: Fluid Queues and the Superposition of ON/OFF Sources with Unequal Characteristics

We consider superpositions of a large number of markovian ON/OFF sources with unequal characteristics and examine the types of limit processes obtained. We also examine multidimensional versions of such processes as inputs to coupled queueing systems and obtain logarithmic asympotics for the workload in the light-tailed case.

### Andrew Richards: Subexponential Geometric Sums. How good are the asymptotics?

Geometric sums of random variables are objects of interest in their own right, and useful in applications. In the case of subexponetial summands, the asymptotics of the tail distribution of the sum is easy to obtain, but finding an upper bound for the sum is a hard problem. We consider one method for constructing an upper bound for the relative error of the asymptotic relation, and show how to use the result in practice using several examples.

### Ju-Yi Yen: Asset allocation: with non-Gaussian factors and exponential utilities

We build a multivariate portfolio model and analyze empirical results of the investment. A signal processing technique known as independent component analysis is applied to decompose multivariate financial time series into statistically independent components (ICs). These ICs are assumed to follow the pure jump non-Gaussian process. The process allows one to capture movements in skewness and kurtosis as well as volatility.

### Boris Bocharov: Solving stochastic integral inclusions using maximal monotone operators.

Stochastic integral inclusions with maximal monotone operators are considered in Banach Spaces. Implicit numerical scheme(s) are presented. Furthermore, the convergence of the approximations to the solution of the inclusion is established. Finally, it is suggested how the original problem can be generalized in a number of ways.

### Dmitry Korshunov: Lamperti's approach to transience, positive and null reccurence

The first part of the talk is devoted to classification of one-dimensional Markov chains according to the asymptotic behaviour of the first two moments of their jumps. This classification is due to J. Lamperti, from 60's. The second part of the talk is about heavy-traffic analysis for stationary distributions of Markov chains. Also, stability problems for stationary distributions will be considered. We conclude with applications to ALOHA and to queueing systems with state-dependent service rate.

### Amanda Turner: Planar aggregation and the coalescing Brownian flow

Diffusion limited aggregation (DLA) is a random growth model which was originally introduced in 1981 by Witten and Sander. This model is prevalent in nature and has many applications in the physical sciences as well as industrial processes. Unfortunately it is notoriously difficult to understand, and only one rigorous result has been proved in the last 25 years. We consider a simplified version of the Hastings-Levitov model of planar aggregation which is obtained by composing certain independent random conformal maps and show that the evolution of harmonic measure on the boundary of the cluster converges to the coalescing Brownian flow.

### Vitali Wachtel: Heavy-traffic analysis of the maximum of an asymptotically stable random walk.

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to $1$ and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance.

### Dmitry Korshunov: On problem of stability of Markov chains in infinite-dimensional spaces

On a particular example of a Markovian polling system with infinitely many stations we discuss the problem of stability (ergodicity) of the corresponding infinite-dimensional Markov process of queue lengths. The corresponding Markov chain is not irreducible, as there are clearly states which do not communicate, e.g. the states (0,0,...) and (1,1,...). Then, for the corresponding infinite-dimensional Markov process, the usual type of ergodicity cannot in general occur, and it is necessary to introduce a modified concept, sometimes called weak ergodicity, which corresponds to the convergence of the finite-dimensional distributions of the process. Of particular interest is the identification of functionals of the process which do converge to the stationary version.

### Henrik Hult: Efficient calculation of risk measures by importance sampling - the heavy-tailed case

Computation of extreme quantiles and tail-based risk measures using standard Monte Carlo simulation can be inefficient. A large number of samples are needed to obtain reliable estimates. A method to the reduce computational cost is to use importance sampling. Then samples are generated from a sampling distribution and each sample is weighted according to its likelihood ratio. We show that importance sampling algorithms, designed for efficient tail probability estimation, can significantly improve Monte Carlo estimators of risk measures, such as Value-at-Risk and Expected Shortfall. In the heavy-tailed setting, when the random variable of interest has a regularly varying distribution, we provide sufficient conditions for the asymptotic relative error of importance sampling estimators to be small. The results are illustrated by numerical examples.

### Vladas Sidoravicius: On non-uniqueness for specifications (g-functions or chains with infinite connections)

The question of whether a specification, also referred sometime as g-function or â€œchain with infinite connectionsâ€, uniquely determines the stationary process under natural â€œmixingâ€ assumption, has been prominent since the pioneering work of DÃ¶blin and Fortet (1937). However in the past time reasonable progress was achieved only in providing sufficient conditions for the uniqueness of Gibbs measure. Bramson and Kalikow (1993) provided a remarkable, and for decades unique example of a continuous and regular specification that admits multiple Gibbs measures. During the talk I will review already known facts and will present a new non-uniqueness result, which in combination with uniqueness criteria of Johansson and Ã–berg (2003) determines the optimal modulus of continuity for a specification which admits multiple Gibbs measures.

### Daniel Ueltschi: Random Permutations with Nonuniform Distributions

I will discuss various models of random permutations with nonuniform distributions. One model involves weights that depend on the cycle structure of the permutation. The other models deal with permutations of points in space, and there is an additional weight that involves the length of permutation jumps. The main question is about the possible occurrence of infinite cycles at high enough density. These models of random permutations are motivated by quantum statistical mechanics and by mathematical biology. (This is joint work with V. Betz and Y. Velenik.)

### Seva Shneer: Slotted versus non-slotted CSMA: throughputs and fairness

We consider a system consisting of n transmitters on a line. The channel used by transmitters is such that 2 of them cannot send messages simultaneously if the distance between them is not larger than k, where k is a positive integer. The well-known CSMA protocol is often used to govern the behaviour of the transmitters in this scenario, and the performance of the system in this case is well understood. We consider a discrete-time analogue of the CSMA protocol and in the case of a completely saturated system find exact formulas for the total throughput of the system governed by this algorithm and for individual throughputs of all transmitters. We then compare these throughputs with the throughputs achieved by the classical CSMA protocol. The fairness of the two protocols is also compared. Finally, we discuss bounds for the throughput of a system where the first transmitter (source) is saturated and messages are relayed by the intermediate transmitters to the last one (destination). A number of directions for future research will be discussed at the end of the talk.

### Jerzy Jaworski: Classical Random Mappings and their Applications

We give a short overview of classical random mappings models and give some examples of the basic results. These models have natural applications in cryptology. In particular, questions concerning the structure of random digraphs representing mappings arise in the analysis of cryptographic systems (e.g. DES), in applications of Pollard's algorithm, in simulations of shift register sequences, and in computational number theory and random number generation. To date, the most widely studied models have been special cases of a model whose key feature is that each vertex in the corresponding digraph chooses' the vertex that it is mapped to independently of the choices' made by all other vertices. First we will focus on results for such random mappings with independent choices of the images which are related to some of these applications. Then we will also consider random mappings for which the vertex `choices' are not necessarily independent (the joint work with Jennie Hansen). In particular, a general model is constructed by first specifying the in-degrees of the vertices (using a collection of non-negative integer-valued exchangeable random variables) and then selecting a random mapping uniformly from all mappings with the given in-degree sequence. We focus on the corresponding properties of two special cases - the preferential and anti-preferential attachment models.

### Sandeep Juneja: Estimating Mean of Non-Linear Function of a Conditional Expectation with Applications to Portfolio Risk Measurement

In this talk we consider the estimation of E(G[ E(Y|X)]) where G is a non linear function, Y takes real values while X may be multi-dimensional. Applications include risk measurement of a portfolio containing derivatives where the value of a portfolio at a time horizon of interest is a conditional expectation that depends on the value of risk factors at that time. This estimation requires nested simulation: In the outer loop independent samples of X are generated while in the inner loop for each generated X, many samples of Y are generated. Practitioners often view this methodology to be computationally prohibitive and develop ad-hoc approximations that may be inaccurate. In this talk, we analyze the trade-off between outer and inner step computational allocation and identify the rate of convergence of the mean square error of the optimal allocation scheme as the computational effort increases to infinity. We note that for large portfolios it may be optimal to simply generate a single sample in the inner loop thus suggesting that optimally done, estimation via accurate models is computationally viable. We also compare the nested simulation with alternate Kernel based methods for and show that the latter may perform better when the underlying random process has low dimensionality, while pure nested simulation may be preferred otherwise.

### Sandeep Juneja: The Concert Queuing Game: To Wait or To be Late

We introduce the concert (or cafeteria) queueing problem: A finite but large number of customers arrive into a queueing system that starts service at a specified opening time. Each customer is free to choose her arrival time (before or after opening time), and is interested in early service completion with minimal wait. These goals are captured by a cost function which is additive and linear in the waiting time and service completion time, with coefficients that may be class dependent. We analyze the system in the many-customer asymptotic regime and develop a fluid limit for the resulting queueing system. We consider a fluid model of this system, which is motivated as the fluid-scale limit of the stochastic system. In the fluid setting, we explicitly identify the unique Nash-equilibrium arrival profile for each class of customers. Our structural results imply that, in equilibrium, the arrival rate is increasing up until the closing time where all customers are served. Furthermore, the waiting queue is maximal at the opening time, and monotonically decreases thereafter. In the simple single class setting, we show that the price of anarchy (PoA, the efficiency loss relative to the socially optimal solution) is exactly two, while in the multi-class setting we develop tight upper and lower bounds on the PoA. In addition, we consider several mechanisms that may be used to reduce the PoA. We also discuss Nash equilibrium in the single class finite customer setting and show its convergence to the fluid limit equilibrium. The proposed model may explain queueing phenomena in diverse settings that involve a pre-assigned opening time.

### Andrew Wade: Non-homogeneous random walks

For this talk a random walk is a discrete-time time-homogeneous Markov process on d-dimensional Euclidean space. If such a random walk is spatially homogeneous, its position can be expressed as a sum of independent identically distributed random vectors, and these homogeneous random walks are well understood. The most subtle case is when the mean dirft (i.e., average increment) of the walk is zero.

The assumption of spatial homogeneity, while simplifying the mathematical analysis, is not always realistic for applications. As soon as the spatial homogeneity assumption is relaxed, the situation becomes much more complicated: a non-homogeneous random walk can be transient in two dimensions, for instance.

I will give an introduction to some results on non-homogeneous random walks with asymptotically zero mean-drift, that is, the magnitude of the drift at a point tends to 0 as the distance of that point from the origin tends to infinity. It turns out that this is the natural regime in which to look for important phase transitions in asymptotic behaviour. This includes work by Lamperti in the 1960s on recurrence/transience behaviour.

I will also discuss recent joint work with Iain MacPhee and Mikhail Menshikov (Durham) concerned with angular asymptotics, i.e., exit-from-cones problems. We show that, in contrast to recurrence/transience behaviour, the angular properties of non-homogeneous random walks are remarkably well-behaved in some sense in the asymptotically zero drift regime.

### Terence Chan: Ruin probability asymptotics and bounds with investment of surplus

In this talk I will consider asymptotics (as the initial capital goes to infinity) and upper bounds of the ruin probability of a risk process with a spectrally positive Levy process as claims and constant premium rate, where the surplus is invested. Results are obtained for thin- and fat-tailed claims distributions and investment in riskless and risky assets.

### Dimitrios Konstantinides: Precise large deviations for sums of negatively dependent random variables with common long tailed distributions

We investigate the precise large deviations for negatively dependent random variables. We prove general asymptotic relations for both the partial sums S_n for the long tailed distributions and the random sums S_{N_t} for the subexponential distributions, where the N_t is an integer counting process. It is found out that the precise large deviations for negatively dependent random variables are insensitive to this kind of dependence. Finally, we present applications on the classical counting processes, Poisson and renewal.

### Jonathan Jordan: Geometric preferential attachment random graphs

Preferential attachment (or "scale-free") random graphs, in which a growing network develops by new vertices attaching preferentially to existing vertices which already have a high degree, were proposed, originally by Barabási and Albert, as models for networks appearing in a wide range of contexts (including biological, technological and social) in which examination of data often reveals an approximately power law distribution of vertex degrees. It was rigorously shown by Bollobás at al that preferential attachment graphs did indeed have this property. In many of the contexts in which random graph models are used it makes sense for the vertices to have some location in space. The original preferential attachment model has no spatial element, and in this talk I will describe a model which combines a preferential attachment element with a spatial element. I will describe results which show that under certain conditions on the spatial element the power law degree property is retained, and discuss what may happen when these conditions do not apply.

### Guenter Last: On the chaos expansion on general Poisson spaces

We establish an explicit Wiener-Ito chaos expansion of Poisson functionals. We also present some consequences for the variational calculus of Poisson processes. Parts of the talk are based on joint work with Mathew Penrose (Bath).

### Neil Walton: Insensitive, maximum stable allocations converge to proportional fairness

We describe a queueing model where service is allocated as a function of queue sizes. We consider allocations policies that are insensitive to service requirements and have a maximal stability region. We take a limit where the queueing model become congested. We study how service is allocated under this limit. We demonstrates that the only possible limit allocation is one that maximizes a proportionally fair optimization problem.

### Michel Mandjes: Birthday Surprises

The classical birthday problem is a standard exercise for our students: what is the probability that, among k people, everyone has a different birthday? Things complicate enormously, however, in the situation that the birthdays are not equally spread over the year, and also other ramifications can be thought of. In this presentation I will look at various generalized birthday problems, in various asymptotic regimes. This talk contains joint work with Sandeep Juneja (TIFR)

### Damon Wischik: What mathematicians should know about the design principles of the Internet

The IETF will soon sign off an Experimental Standard for multipath TCP. This is the fruit of four years of work by a group of us in the computer science department at UCL, aimed at implementing the maths in a 2005 paper by Kelly and Voice, which was prompted by a 2003 paper by Han, Shakkottai et al. I will describe the repositionings we made en route to deployment, and discuss which aspects of the maths are in tune with the design principles of the Internet.