10-11 September 2015
Heriot-Watt University, Colin Maclaurin building, room CM.S01. For details of how to get to Heriot-Watt University, click here. To find out how to get to Colin Maclaurin building, this map can also be useful.
Lunch will be provided. There will also be organised coffee breaks on both days.
| 09.20 - 09.30 | Opening |
| 09.30 - 10.30 | Soren Asmussen, Maxwell Lecturer |
| Markov Renewal Methods in Restart Problems in Complex Systems | |
| A task with ideal execution time $L$ such as the execution of a computer program or the transmission of a file on a data link may fail, and the task then needs to be restarted. The task is handled by a complex system with features similar to the ones in classical reliability: failures may be mitigated by using server redundancy in parallel or $k$-out-of-$n$ arrangements, standbys may be cold or warm, one or more repairmen may take care of failed components, etc. The total task time $X$ (including restarts and pauses in failed states) is investigated with particular emphasis on the tail $\mathbb{P}(X>x)$. A general alternating Markov renewal model is proposed and an asymptotic exponential form $\mathbb{P}(X>x)\sim C\mathrm{e}^{-\gamma x}$ identified for the case of a deterministic task time $L\equiv \ell$. The rate $\gamma$ is given by equating the spectral radius of a certain matrix to 1, and the asymptotic form of $\gamma=\gamma(\ell)$ as $\ell\to\infty$ is derived, leading to the asymptotics of $\mathbb{P}(X>x)$ for random task times $L$. A main finding is that $X$ is always heavy-tailed if $L$ has unbounded support. The case where the Markov renewal model is derived by lumping in a continuous-time finite Markov process with exponential holding times is given special attention, and the study includes analysis of the effect of processing rates that differ with state or time. We also consider systems with time-inhomogeneous Poisson failures. | |
| 10.30 - 11.00 | Coffee break |
| 11.00 - 11.45 | Alexander Sakhanenko |
| On a limiting behaviour of a conditional random walk with bounded local times | |
| We consider a random walk on the integers with i.i.d. jumps taking value 1 and negative values, and with a limited number of visits, say $L$, to each state. The latter means that the walk stops (``freezes'') at any state if it visits the state the $(L+1)$st time. Such a walk freezes at some state with probability one and a probability to hit a large level, say $N$, tends to zero when $N$ grows to infinity. We analyse asymptotic properties of the trajectory up to the hitting time of level $N$ given that the hitting time is finite. Itai Benjamini and Nathana\"el Berestycki (2010) considered the symmetric simple random walk and showed, in particular, that the limiting process has a regenerative structure. We generalise their results using different techniques. We will discuss further a number of extensions of the model. The talk is based on a joint work with Sergey Foss. | |
| 12.00 - 12.45 | Denis Denisov |
| 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 | |
| 13.00-14.00 | Lunch |
| 14.00 - 14.45 | Vitali Wachtel |
| Invariance principles for random walks in cones | |
| We discuss invariance principles for conditioned random walks. In the first part of the talk we review known results for one-dimensional random walks conditioned to stay positive. And in the second part we present our results for mulditimensional random walks conditioned to stay in a cone. The talk is based on a joint work with Jetlir Duraj. | |
| 15.00 - 15.45 | Dima Korshunov |
| 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 Cram\'er'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. | |
| 19.30 | Conference dinner (invitation only) |
| 09.30 - 10.15 | Takis Konstantopoulos |
| Burke's theorem and Brownian motion | |
| In this talk I will review a nonlinear transformation of two independent Brownian motions that results in a Brownian motion. Burke's theorem and Skorokhod reflection is what is responsible for this. | |
| 10.30 - 11.00 | Coffee break |
| 11.00 - 11.45 | Anatolii Mogulskii |
| Large deviations for processes with independent increments | |
| The talk is devoted to the large deviation principles for processes with independent increments. The results include the so-called local and extended large deviation principles that hold in those cases where the ``usual'' (classical) large deviation principle is inapplicable. | |
| 12.00 - 12.45 | Kostya Borovkov |
| Continuity Problems for Boundary Crossing Probabilities | |
| Computing the probability for a given diffusion process to stay under a particular boundary is crucial in many important applications including pricing financial barrier options. It is a rather tedious task that, in the general case, requires the use of some approximation methodology. One possible approach to this problem is to approximate given (general curvilinear) boundaries with some other boundaries, of a form enabling one to relatively easily compute the boundary crossing probability. We discuss results on the accuracy of such approximations for both the Brownian motion process and general time-homogeneous diffusions, their extensions to the multivariate case, and also some contiguous topics. | |
| 13.00 - 14.15 | Lunch |
| 14.15 - 15.00 | George Delligiannidis |
| Variance of partial sums of stationary processes | |
| We give necessary and sufficient conditions for the variance of the partial sums of stationary processes to be regularly varying in terms of the spectral measure associated with the shift operator. In the case of reversible Markov chains, or Markov chains with normal transition operator we also give necessary and sufficient conditions in terms of the spectral measure of the transition operator. The two spectral measures are then linked through the use of harmonic measure. This is joint work with S. Utev(University of Leicester, UK) and M. Peligrad (University of Cincinnati, USA). | |
| 15.15 - 16.00 | Stan Zachary |
| Optimization of energy storage in power networks | |
| Electric power networks require that supply and demand be kept balanced on a minute-by-minute basis. Storage, which may be regarded as shifting energy through time, is able to assist in this process by (a) smoothing predicted imbalances, such as between known times of high and low demand, (b) buffering against sudden and unpredicted variations in either supply or demand. Typically the economic use of storage requires that it is simultaneously able to provide both these (and often also other) services. We discuss an integrated mathematical framework for optimal control of such storage and for the determination of its economic value. We discuss also possible extensions to the time-shifting of demand. |
Attendance at the talks is free and there is no need to register. However, it would help us in planning the event if you send an email to Seva Shneer indicating your interest in attending.
The organisers gratefully acknowledge financial support from the Maxwell Institute.