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Applied Probability and Operations Research
Stochastic processes are commonly used to model a wide-range of physical processes; for example stock prices, queues, evolution of genes in a population etc. It is important to learn, both through theoretical results and simulation, about properties of stochastic processes in order to control and make predictions about the real-life processes that they model.
The applied probability group consists of Kevin Glazebrook, Amanda Turner, Paul Fearnhead and Chris Sherlock along with various students and postdocs. The group's interests lie in the simulation, inference and control of stochastic processes in general. Of particular current interest are diffusion processes and Markov chains. Research on the (dynamic) control of random systems focusses on the development of effective and computationally implementable alternatives to stochastic dynamic programming.
Control of Stochastic Processes
Many situations arise where decisions which impact the evolution of some complex randomly evolving system need to be made in real time. One application of current interest concerns the optimal management of a network of retail depots which have the capacity to supply each other (transhipment) when a demand arises at a location which cannot meet it from stock already held. The current state of such a system would typically be a vector of high dimension describing inventory levels at each location. The classic approach to the development of optimal policies in such contexts is (stochastic) dynamic programming (DP). Sadly, straightforward application of DP in systems of any complexity is computationally infeasible and effective alternatives are required. If such alternatives produce policies whose cost performance can be demonstrated to be close to optimal, so much the better. Current research focusses on the utilisation of Lagrangian relaxation, policy improvement and state-space partitioning to develop effective policies for classes of stochastic decision processes which model resource allocation problems.
Scaling limits of stochastic processes
The simplest example of a scaling limit of a random process is the law of large numbers for random variables which states that the average value of a sequence of independent identically distributed random variables converges to a deterministic limit as the length of the sequence tends to infinity. More generally one can ask what happens to the behaviour of a random process whose jump sizes tend to zero as the jump rate tends to infinity. Such processes arise in a variety of contexts, in particular in physical settings where the process describes the random molecular behaviour on a microscopic level, and the existence of scaling limits corresponds to stable behaviour on a macroscopic level. The molecular behaviour can be modelled as a stochastic jump process, where the jump sizes are proportional to atomic lattice spacing or particle sizes, or inversely proportional to the number of particles. Diffusions
An important class of stochastic processes are diffusions, which model processes that evolve continuously over continuous time. Whilst widely used, particularly to model financial time series, until recently it had been impossible to even simulate exactly from many diffusion models; and various approximate method were required to perform inference for diffusions. A major breakthrough by the members of the applied probability group at Lancaster, has led to computationally efficient algorithms for exact simulation of, and inference for, a much wider class of diffusions. Part of this work was recently presented as a paper read to the Royal Statistical Society.
Simulation of Diffusions
A common method for simulating a random variable from an intractable density, is to simulate realisations from a different density and accept or reject this realisation with a probability proportional to the ratio of the two densities. This idea, called rejection sampling, can be extended to allow simulation from an intractable diffusion, by simulating a realisation of Brownian motion and applying an accept-reject step. However, the decision whether to accept the proposed Brownian motion, is made tricky by the fact that you cannot calculate the probability with which you should accept it! Recent work by the group has shown how the decision of whether to accept the path can be made (correctly) despite this. Ongoing research is extending these methods to an ever-increasingly wider class of diffusions.
Inference for Diffusions
Likelihoods for partially or discretely observed diffusions are in general intractable. However the methods developed for simulating from such diffusions enable Monte Carlo estimates of the likelihood to be generated. These estimates are exact in the limit of the Monte Carlo sample size tending to infinity.
