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Univariate marked Hawkes processes are used to model a range of real-world phenomena including earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, and order book dynamics. This paper illustrates a fundamental connection between univariate marked Hawkes processes and multivariate Hawkes processes. Exploiting this connection renders a framework that can be built upon for expressive and flexible inference on diverse data. Specifically, multivariate unmarked Hawkes representations are introduced as a tool to parameterize univariate marked Hawkes processes. We show that such multivariate representations can asymptotically approximate a large class of univariate marked Hawkes processes, are stationary given the approximated process is stationary, and that resultant conditional intensity parameters are identifiable. A simulation study demonstrates the efficacy of this approach, and provides heuristic bounds for error induced by the relatively larger parameter space of multivariate Hawkes processes., Comment: 26 pages, 3 figures, submitted to the Annals of Statistics

Most point process models for earthquakes currently in the literature assume the magnitude distribution is i.i.d. potentially hindering the ability of the model to describe the main features of data sets containing multiple earthquake mainshock aftershock sequences in succession. This study presents a novel multidimensional fractional Hawkes process model designed to capture magnitude dependent triggering behaviour by incorporating history dependence into the magnitude distribution. This is done by discretising the magnitude range into disjoint intervals and modelling events with magnitude in these ranges as the subprocesses of a mutually exciting Hawkes process using the Mittag-Leffler density as the kernel function. We demonstrate this model's use by applying it to two data sets, Japan and the Middle America Trench, both containing multiple mainshock aftershock sequences and compare it to the existing ETAS model by using information criteria, residual diagnostics and retrospective prediction performance. We find that for both data sets all metrics indicate that the multidimensional fractional Hawkes process performs favourably against the ETAS model. Furthermore, using the multidimensional fractional Hawkes process we are able to infer characteristics of the data sets that are consistent with results currently in the literature and that cannot be found by using the ETAS model., Comment: 37 pages, 10 tables, 3 figures

This paper extends the existing fractional Hawkes process to better model mainshock-aftershock sequences of earthquakes. The fractional Hawkes process is a self-exciting point process model with temporal decay kernel being a Mittag-Leffler function. A maximum likelihood estimation scheme is developed and its consistency is checked. It is then compared to the ETAS model on three earthquake sequences in Southern California. The fractional Hawkes process performs favourably against the ETAS model. Additionally, two parameters in the fractional Hawkes process may have a fixed geophysical meaning dependent on the study zone and the stage of the seismic cycle the zone is in., Comment: 16 pages, 7 figure, submitted to the Journal of the Royal Statistical Society Series C

This paper describes the development and efficacy of an online tool for assessing the numeracy of undergraduate students. The tool was designed to be easy to administer, provide immediate feedback to students on whether they had the required level of numeracy, and to be consistent with other measures of adult numeracy. When used with students taking a mathematics or statistics course, we found a significant correlation of r = 0.45 between their numeracy score and final mark in their enrolled course. Students who had a numeracy score less than our threshold had a 30.6% probability of failing their course, whereas students who had a numeracy score of at least our threshold had a probability of failing of only 8.0%.

We develop a finite difference approximation of order $\alpha$ for the $\alpha$-fractional derivative. The weights of the approximation scheme have the same rate-matrix type properties as the popular Gr\"unwald scheme. In particular, approximate solutions to fractional diffusion equations preserve positivity. Furthermore, for the approximation of the solution to the skewed fractional heat equation on a bounded domain the new approximation scheme keeps its order $\alpha$ whereas the order of the Gr\"unwald scheme reduces to order $\alpha-1$, contradicting the convergence rate results by Meerschaert and Tadjeran.

We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided L\'evy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided L\'evy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified L\'evy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified L\'evy process by showing continuity of the modifications with respect to the Skorokhod topology., Comment: 41 pages, submitted for publication

We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided L\'evy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided L\'evy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified L\'evy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified L\'evy process by showing continuity of the modifications with respect to the Skorokhod topology., Comment: 81 pages, submitted for publication

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\Omega)$ and $L_1(\Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.

This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well-posed Dirichlet problem for space-time fractional diffusions in one or several variables. This solves an open problem in numerical analysis., Comment: 26 pages, 1 figure

It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory., Comment: in Proceedings of the American Mathematical Society, Published electronically July 12, 2016

We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-$p$ H\"older continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-H\"older) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.

Numerical transport models based on the advection-dispersion equation (ADE) are built on the assumption that sub-grid cell transport is Fickian such that dispersive spreading around the average velocity is symmetric and without significant tailing on the front edge of a solute plume. However, anomalous diffusion in the form of super-diffusion due to preferential pathways in an aquifer has been observed in field data, challenging the assumption of Fickian dispersion at the local scale. This study develops a fully Lagrangian method to simulate sub-grid super-diffusion in a multi-dimensional regional-scale transport. The underlying concept is based on previous observations that solutions to space-fractional ADEs, which can describe super-diffusive dispersion, can be obtained by transforming solutions of classical ADEs. The transformations are equivalent to randomizing particle travel time or relative velocity for each model time step. Here, the time randomizing procedure known as subordination is applied to flow field output from MODFLOW simulations. Numerical tests check the applicability of the novel method in mapping regional-scale super-diffusive transport conditioned on local properties of multi-dimensional heterogeneous media.

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

We give stability and consistency results for higher order Gr\"unwald-type formulae used in the approximation of solutions to fractional-in-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier multipliers to gain regularity and stability results. We then generalise the theory to the case where the first derivative operator is replaced by the generator of a bounded group on an arbitrary Banach space., Comment: To appear in Transactions of the AMS

Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional derivatives in space, and power law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable L\'evy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index $1<\alpha<2$ to the density of the hitting time of a stable subordinator with index $1/\alpha$, and thereby unify some recent results in the literature. These results also provide a concrete interpretation of Zolotarev duality in terms of the fractional diffusion model., Comment: 16 pages

In this paper we develop two mathematical models to predict the release kinetics of a water soluble drug from a polymer/excipient matrix tablet. The first of our models consists of a random walk on a weighted graph, where the vertices of the graph represent particles of drug, excipient and polymer, respectively. The graph itself is the contact graph of a multidisperse random sphere packing. The second model describes the dissolution and the subsequent diffusion of the active drug out of a porous matrix using a system of partial differential equations. The predictions of both models show good qualitative agreement with experimental release curves. The models will provide tools for designing better controlled release devices., Comment: 17 pages, 7 figures; Elaborated at the first Workshop on the Application of Mathematics to Problems in Biomedicine, December 17-19, 2007 at the University of Otago in Dunedin, New Zealand

A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations., Comment: 18 pages, minor spelling corrections