François Baccelli (University of Texas at Austin)

Dynamics on Unimodular Random Networks

The talk will be focused on dynamics on discrete random structures.

We will first consider compatible dynamics on the points of a stationary point process, namely ``rules to navigate from point to point'' which are preserved by translations.

Each such dynamics defines a random graph on the points of the process. The connected components of this graph can be split into a collection of foils, which are the analogue of the stable manifold of the dynamics.

We will give a general classification of such dynamics in terms of the cardinality of the foils of these connected components. There are three types: F/F (finitely many finite foils), I/F (infinitely many finite foils), and I/I.

We will then consider compatible dynamics on random graphs and random networks (marked graphs), namely ``rules to navigate from node to node'' which are preserved by graph or network isomorphisms. We will show that this classification also holds for all unimodular random graphs and networks.

We will complement this by results on the relative intensities of the foils.

These results will be illustrated by concrete examples of navigation rules, both on point processes and on random networks.

Frank Kelly (University of Cambridge)

A Markov Model of a Limit Order Book: Thresholds, Recurrence and Trading Strategies

In this talk we discuss an analytically tractable model of a limit order book where the dynamics are driven by stochastic fluctuations between supply and demand. The model has a natural interpretation for a highly traded market on short time-scales where there is a separation between the time-scale of trading, represented in the model, and a longer time-scale on which fundamentals change.

We describe our main result for the model, which is the existence of an explicit limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. Fluid limits play an important role in establishing the recurrence properties of the model. We use the model to analyse various high-frequency trading strategies (for example market-making, sniping and mixtures of these), and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions.

Takis Konstantopoulos (Uppsala University)

Compositions of Empirical Distribution Functions and Bernstein Operators

We present a stochastic approach, based on compositions of empirical distribution functions, to proving theorems regarding convergence and rates of convergence of iterates of Bernstein operators. This completes the classical theorem of Bernstein for polynomial approximations to continuous functions by relating it to the Wright-Fisher Markov chain and its corresponding diffusion limit.

Hermann Thorisson (University of Iceland)

Finding Patterns in Brownian Motion

We consider the problem of finding particular patterns in a realisation of a two-sided standard Brownian motion. Examples include two-sided Skorohod imbedding, the Brownian bridge and several other patterns, also in planar Brownian motion. The key tool here are recent allocation results in Palm theory.

Sergey Foss (Heriot-Watt University)

Heavy Tails in Generalised Jackson Networks

A probability distribution on the positive half-line is heavy-tailed if it does not have any exponential moment. We are interested in the tail asymptotics for a waiting/sojourn time of a typical customer in a stochastic network driven by service times with heavy-tailed distributions. In a simple setting of a single-server queue, the so-called "principle of a single big jump" holds: the waiting time is large if one of the past service times is large. This is true in the whole diapason of heavy-tailed distribution tails, from regularly varying via log-normal to Weibull with parameter less than 1. The same principle holds in a class of so-called max-plus networks. However, for a more general class of feedforward networks only partial results (for regularly-varying tails) are known. Even less in known for networks where there is a chance to revisit a server.