Owen Jones

Convergence properties of the cross-entropy method for discrete optimization

We present new theoretical convergence results on the cross-entropy (CE) method for discrete optimization. We show that a popular implementation of the method converges, and finds an optimal solution with probability arbitrarily close to 1. We also give conditions under which an optimal solution is generated eventually with probability 1.

Transition probabilities for the simple random walk on the Sierpinski graph

Non-Gaussian upper and lower bounds are obtained for the transition probabilities of the simple random walk on the Sierpinski graph, the pre-fractal associated with the Sierpinski gasket. They are of the same form as bounds previously obtained for the transition density of Brownian motion on the Sierpinski gasket, subject to a scale restriction. A comparison with transition density bounds for random walks on general graphs demonstrates that this restriction represents the scale at which the pre-fractal graph starts to look like the fractal gasket.

Estimating the Hurst index of a self-similar process via the crossing tree

By considering the crossings of size 2/sup n/ made by a continuous self-similar process, for n = ..., -1, 0, 1, ..., we build a tree of crossings that encodes the sample path. From the crossing tree, we obtain a test for self-similarity and an estimator for the Hurst index, H. The accuracy of the estimator is established theoretically and numerically, and a comparison study with three other estimators is given, demonstrating its superior performance. The estimator and self-similarity test are applied to network packet traces, revealing some previously unobserved features.

Exploring Decision Makers Use of Price Information in a Speculative Market

We explore the extent to which the decisions of participants in a speculative market effectively account for information contained in prices and price movements. The horse race betting market is an ideal environment to explore these issues. A conditional logit model is constructed to determine winning probabilities based on bookmakers closing prices and the time-indexed movement of prices to the market close. We incorporate a technique for extracting predictors from price (odds) curves using orthogonal polynomials. The results indicate that closing prices do not fully incorporate market price information, particularly information that is less readily discernable by market participants.

Thick and thin points for random recursive fractals

We consider random recursive fractals and prove fine results about their local behaviour. We show that for a class of random recursive fractals the usual multifractal spectrum is trivial in that all points have the same local dimension. However, by examining the local behaviour of the measure at typical points in the set, we establish the size of fine fluctuations in the measure. The results are proved using a large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.

Simulation of Brownian motion at first-passage times

We show how to simulate Brownian motion not on a regular time grid, but on a regular spatial grid. That is, when it first hits points in @dZ for some @d>0. Central to our method is an algorithm for the exact simulation of @t, the first time Brownian motion hits +/-1. This work is motivated by boundary hitting problems for time-changed Brownian motion, such as appear in mathematical finance when pricing barrier-options.

Modelling the effects of fire and rainfall regimes on extreme erosion events in forested landscapes

Existing models of post-fire erosion have focused primarily on using empirical or deterministic approaches to predict the magnitude of response from catchments given some initial rainfall and burn conditions. These models are concerned with reducing uncertainties associated with hydro-geomorphic transfer processes and typically operate at event timescales. There have been relatively few attempts at modelling the stochastic interplay between fire disturbance and rainfall as factors which determine the frequency and severity with which catchments are conditioned (or primed) for a hazardous event. This process is sensitive to non-stationarity in fire and rainfall regime parameters and therefore suitable for evaluating the effects of climate change and strategic fire management on hydro-geomorphic hazards from burnt areas. In this paper we ask the question, “What is the first-order effect of climate change on the interaction between fire disturbance and storms?” The aim is to isolate the effects of fire and rainfall regimes on the frequency of extreme erosion events. Fire disturbance and storms are represented as independent stochastic processes with properties of spatial extent, temporal duration, and frequency of occurrence, and used in a germ–grain model to quantify the annual area affected by extreme erosion events due to the intersection of fire disturbance and storms. The model indicates that the frequency of extreme erosion events will increase as a result of climate change, although regions with frequent storms were most sensitive.

A CLASS OF MULTIFRACTAL PROCESSES CONSTRUCTED USING AN EMBEDDED BRANCHING PROCESS

We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step n, we can generate step n+ 1 in O(logn) operations. Detailed pseudo-code for this algorithm is provided.

Multifractal spectra for random self-similar measures via branching processes

Start with a compact set K ⊂ R d . This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in R d and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

Asymptotically One-Dimensional Diffusion on the Sierpinski Gasket and Multi-Type Branching Processes with Varying Environment

Asymptotically one-dimensional diffusions on the Sierpinski gasket constitute a one parameter family of processes with significantly different behaviour to the Brownian motion. Due to homogenization effects they behave globally like the Brownian motion, yet locally they have a preferred direction of motion. We calculate the spectral dimension for these processes and obtain short time heat kernel estimates in the Euclidean metric. The results are derived using branching process techniques, and we give estimates for the left tail of the limiting distribution for a supercritical multi-type branching process with varying environment.