Maxwell Institute Probability Day
13th May 2016

The Maxwell Institute Probability Day is an annual event organised jointly between the probability and stochastic analysis groups at
University of Edinburgh and HeriotWatt University. The event aims at fostering interactions between these
research groups and neighbouring universities in the UK.
 Organisers.
 Sergey Foss (Heriot Watt)
 Istvan Gyongy (University of Edinburgh)
 Michela Ottobre (Heriot Watt)
 David Siska (University of Edinburgh)
 Venue.
Lecture Theatre 183, Old College, 68 South Bridge, Edinburgh EH8 9YL
 Registration.
The event is open to all. For catering purposes, please fill
out this
doodle poll by the 5th of May if you would like to attend.
We strongly encourage participation of PhD students, Postdocs and young researchers.
 Programme.
 12.0012.45 Lunch
 12.45  13.30 Tusheng Zhang (Manchester) Lattice Approximations of Reflected Stochastic Partial
Differential Equations Driven by SpaceTime White Noise
 13.30  14.15 Dominic Breit (Heriot Watt) Numerical approximation of stochastic pLaplace type system
 14.15  14.40 Timofei Prasolov (Heriot Watt) Tail asymptotics of meeting times of dependent agents
 14.40  15.10 Coffe Break
 15.10  15.55 Tolga Tezcan (London Business School) Yardstick Competition for Service Systems
 15.55  16.40 Ruth King (University of Edinburgh) Fitting statespace models to data using an approximate hidden Markov model
 Abstracts
 Tusheng Zhang (Manchester) Lattice Approximations of Reflected Stochastic Partial Differential Equations Driven by SpaceTime
White Noise
We introduce a discretization/approximation scheme for reflected
stochastic partial differential equations driven by spacetime white noise through
systems of reflecting stochastic differential equations.
To establish the convergence of the scheme, we study the existence and uniqueness of solutions of Skorohodtype deterministic systems
on timedependent domains. We also need to establish the convergence of an approximation scheme for deterministic parabolic obstacle problems.
 Dominic Breit (Heriot Watt) Numerical approximation of stochastic pLaplace type system.
We consider systems of stochastic evolutionary equations of the pLaplace
type. We study finite element spacetime discretization and prove convergence together with rates.
The approximation error is measured with the quasi norm.
 Timofei Prasolov (Heriot Watt) Tail asymptotics of meeting
times of dependent agents
Motivated by work of N. Litvak and P. Robert "A scaling analysis of a
Cat and Mouse Markov chain", we started to analyse more general object, working name for which is "Dog, Cat and Mouse Markov chain". Here coordinate of "Dog" is a simple random walk in integer space. Movement of "Cat" depends on "Dog" and movement of "Mouse" depends on "Cat". We analyse the position of this system after a long run by finding tail asymptotics of meeting times of all three agents.
 Tolga Tezcan (London Business School) Yardstick Competition for Service Systems
Yardstick competition is a regulatory scheme for local monopolists (e.g., hospitals), where the monopolistsâ€™ reimbursement is linked to its performance relative to other equivalent monopolists. This regulatory scheme is known to work well in providing costreduction incentives and offers the theoretical underpinning behind the hospital prospective reimbursement system used throughout the developed world. This paper investigates how yardstick competition performs in service systems (e.g., hospital emergency departments), where in addition to incentivizing cost reduction, the regulator's goal is to provide incentives to reduce customer waiting times. We show that i) the form of yardstick competition used in practice results in inefficiently long waiting times; ii) yardstick competition can be appropriately modified to achieve the dual goal of cost and waitingtime reduction, and present several extensions that help guide on how it could be used in practice.
 Ruth King (University of Edinburgh) Fitting statespace models to data using an approximate hidden Markov
model
Many mathematical models can be expressed in the form of a statespace model. For such models the model can be
separated into two parts: (i) system process  describing the evolution of the true underlying process over
time (assumed to be continuous); and (ii) observation process  describing what we observe given the true state of the process. Typically we wish to obtain inference on the underlying system process, estimating the associated model parameters to further our understanding of the system. However, the corresponding likelihood of the observed data is typically analytically intractable. Estimates of the parameters can be obtained using the Kalman filter assuming a linear Gaussian model. We discuss an alternative approach that can be generally applied to nonlinear and nonGaussian statespace models using an approximate (discretised) hidden Markov model likelihood framework.
The organizers gratefully acknowledge financial support from the Maxwell Institute