In this article, local limit theorems for sequences of simple random walks on graphs are establis... more In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.
We consider the random walk on supercritical percolation clusters in Z d . Previous papers have o... more We consider the random walk on supercritical percolation clusters in Z d . Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.
Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which... more Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. In this paper we examine a simple spot price model that is the exponential of the sum of an Ornstein-Uhlenbeck and an independent mean reverting pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of the spot price process at maturity T . Hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulae for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process.
We establish the asymptotic behaviour of the partition function, the heat content, the integrated... more We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form x γ φ(log x) for a suitable exponent γ and φ a periodic function. We also discuss similar results for the heat content of affine nested fractals.
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that... more We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced. The set of these paths is the Reduced Path Group. It is a continuous analogue of the group of reduced words. The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.
We use the random self-similarity of the continuum random tree to show that it is homeomorphic to... more We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application we determine the mean and almostsure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form. *
Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of... more Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric α-stable processes are used to construct new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from these systems and this enables us to deduce various limiting distribution results for the fluctuations of the associated counting functions. In particular, knowledge of the number variance allows us to introduce and characterize a novel family of centered, long memory Gaussian processes. We obtain fractional Brownian motion as a weak limit of these constructed processes.
On a large class of p.c.f. (finitely ramified) self-similar fractals with possibly little symmetr... more On a large class of p.c.f. (finitely ramified) self-similar fractals with possibly little symmetry we consider the question of existence and uniqueness of a Laplace operator. By considering positive refinement weights (local scaling factors) which are not necessarily equal we show that for each such fractal, under a certain condition, there are corresponding refinement weights which support a unique self-similar Dirichlet form. As compared to previous results, our technique allows us to replace symmetry by connectivity arguments.
In this paper a string is a sequence of positive non-increasing real numbers which sums to one. F... more In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.
We consider a sequence of random graphs constructed by a hierarchical procedure. The construction... more We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p and 1−p respectively. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at p = 1 2 .
We study two-sided heat kernel estimates on a class of fractal graphs which arise from a subclass... more We study two-sided heat kernel estimates on a class of fractal graphs which arise from a subclass of finitely ramified fractals. These fractal graphs do not have spatial symmetry in general, and we find that there is a dependence on direction in the estimates. We will give a new form of expression for the heat kernel estimates using a family of functions which can be thought of as a "distance for each direction". As an application, we give a law of the iterated logarithm which shows that the directional dependence leads to non-uniform behaviour in the typical paths of the random walk.
We show that under a 3 + δ moment condition (where δ > 0) there exists a 'Hartman-Winter' Law of ... more We show that under a 3 + δ moment condition (where δ > 0) there exists a 'Hartman-Winter' Law of the Iterated Logarithm for random walks conditioned to stay non-negative. We also show that under a second moment assumption the conditioned random walk eventually grows faster than n 1/2 (log n) −(1+ε) for any ε > 0 and yet slower than n 1/2 (log n) −1 . The results are proved using three key facts about conditioned random walks. The first is the relation of its step distribution to that of the original random walk given by . The second is the pathwise construction in terms of excursions in and the third is a new Skorohod type embedding of the conditioned process in a Bessel-3 process.
We discuss Monte Carlo methods for valuing options with multiple exercise features in discrete ti... more We discuss Monte Carlo methods for valuing options with multiple exercise features in discrete time. By extending the recently developed duality ideas for American option pricing we show how to obtain estimates on the prices of such options using Monte Carlo techniques.
Abstract We consider an increasing supercritical branching process in a random environ-ment and o... more Abstract We consider an increasing supercritical branching process in a random environ-ment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of ...
When viewed at an appropriate scale, a disordered medium can behave as if it is strictly less tha... more When viewed at an appropriate scale, a disordered medium can behave as if it is strictly less than three dimensional. As fractals typically have non-integer dimensions, they are natural models for disordered media, and di usion on fractals can be used to model transport in disordered media. In particular, such di usion processes can be used to obtain bounds on the fundamental solution to the heat equation on a fractal. In this paper we review the work in this area and describe how bounds on branching processes lead to bounds on heat kernels.
This paper is at an interface between analysis, topology and control theory. In it we prove a non... more This paper is at an interface between analysis, topology and control theory. In it we prove a non-commutative analogue of the result that functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. The equivalence relation determined by the "null sets" for this new problem enable us to construct a continuous analogue of the free group as a quotient subspace of the space of paths of bounded variation. In this development our main theorem extends the work of K.T. Chen, in particular the paper [1], where he considers a similar theorem under a greater smoothness assumption.
In this article, local limit theorems for sequences of simple random walks on graphs are establis... more In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.
We consider the random walk on supercritical percolation clusters in Z d . Previous papers have o... more We consider the random walk on supercritical percolation clusters in Z d . Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.
Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which... more Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. In this paper we examine a simple spot price model that is the exponential of the sum of an Ornstein-Uhlenbeck and an independent mean reverting pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of the spot price process at maturity T . Hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulae for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process.
We establish the asymptotic behaviour of the partition function, the heat content, the integrated... more We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form x γ φ(log x) for a suitable exponent γ and φ a periodic function. We also discuss similar results for the heat content of affine nested fractals.
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that... more We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced. The set of these paths is the Reduced Path Group. It is a continuous analogue of the group of reduced words. The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.
We use the random self-similarity of the continuum random tree to show that it is homeomorphic to... more We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application we determine the mean and almostsure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form. *
Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of... more Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric α-stable processes are used to construct new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from these systems and this enables us to deduce various limiting distribution results for the fluctuations of the associated counting functions. In particular, knowledge of the number variance allows us to introduce and characterize a novel family of centered, long memory Gaussian processes. We obtain fractional Brownian motion as a weak limit of these constructed processes.
On a large class of p.c.f. (finitely ramified) self-similar fractals with possibly little symmetr... more On a large class of p.c.f. (finitely ramified) self-similar fractals with possibly little symmetry we consider the question of existence and uniqueness of a Laplace operator. By considering positive refinement weights (local scaling factors) which are not necessarily equal we show that for each such fractal, under a certain condition, there are corresponding refinement weights which support a unique self-similar Dirichlet form. As compared to previous results, our technique allows us to replace symmetry by connectivity arguments.
In this paper a string is a sequence of positive non-increasing real numbers which sums to one. F... more In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.
We consider a sequence of random graphs constructed by a hierarchical procedure. The construction... more We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p and 1−p respectively. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at p = 1 2 .
We study two-sided heat kernel estimates on a class of fractal graphs which arise from a subclass... more We study two-sided heat kernel estimates on a class of fractal graphs which arise from a subclass of finitely ramified fractals. These fractal graphs do not have spatial symmetry in general, and we find that there is a dependence on direction in the estimates. We will give a new form of expression for the heat kernel estimates using a family of functions which can be thought of as a "distance for each direction". As an application, we give a law of the iterated logarithm which shows that the directional dependence leads to non-uniform behaviour in the typical paths of the random walk.
We show that under a 3 + δ moment condition (where δ > 0) there exists a 'Hartman-Winter' Law of ... more We show that under a 3 + δ moment condition (where δ > 0) there exists a 'Hartman-Winter' Law of the Iterated Logarithm for random walks conditioned to stay non-negative. We also show that under a second moment assumption the conditioned random walk eventually grows faster than n 1/2 (log n) −(1+ε) for any ε > 0 and yet slower than n 1/2 (log n) −1 . The results are proved using three key facts about conditioned random walks. The first is the relation of its step distribution to that of the original random walk given by . The second is the pathwise construction in terms of excursions in and the third is a new Skorohod type embedding of the conditioned process in a Bessel-3 process.
We discuss Monte Carlo methods for valuing options with multiple exercise features in discrete ti... more We discuss Monte Carlo methods for valuing options with multiple exercise features in discrete time. By extending the recently developed duality ideas for American option pricing we show how to obtain estimates on the prices of such options using Monte Carlo techniques.
Abstract We consider an increasing supercritical branching process in a random environ-ment and o... more Abstract We consider an increasing supercritical branching process in a random environ-ment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of ...
When viewed at an appropriate scale, a disordered medium can behave as if it is strictly less tha... more When viewed at an appropriate scale, a disordered medium can behave as if it is strictly less than three dimensional. As fractals typically have non-integer dimensions, they are natural models for disordered media, and di usion on fractals can be used to model transport in disordered media. In particular, such di usion processes can be used to obtain bounds on the fundamental solution to the heat equation on a fractal. In this paper we review the work in this area and describe how bounds on branching processes lead to bounds on heat kernels.
This paper is at an interface between analysis, topology and control theory. In it we prove a non... more This paper is at an interface between analysis, topology and control theory. In it we prove a non-commutative analogue of the result that functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. The equivalence relation determined by the "null sets" for this new problem enable us to construct a continuous analogue of the free group as a quotient subspace of the space of paths of bounded variation. In this development our main theorem extends the work of K.T. Chen, in particular the paper [1], where he considers a similar theorem under a greater smoothness assumption.
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