Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps

AbstractWe consider semi-implicit methods for stochastic age-dependent population equations with Poisson jumps. The main purpose of this paper is to show the convergence of the numerical approximation solution to the true solution with strong order p=12

Geometrically stopped Markovian random growth processes and Pareto tails

Many empirical studies document power law behavior in size distributions of economic interest such as cities, firms, income, and wealth. One mechanism for generating such behavior combines independent and identically distributed Gaussian additive shocks to log-size with a geometric age distribution. We generalize this mechanism by allowing the shocks to be non-Gaussian (but light-tailed) and dependent upon a Markov state variable. Our main results provide sharp bounds on tail probabilities, a simple equation determining Pareto exponents, and comparative statics. We present two applications: we show that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic general equilibrium model with idiosyncratic investment risk are Paretian, and (ii) a random growth model for the population dynamics of Japanese municipalities is consistent with the observed Pareto exponent but only after allowing for Markovian dynamics

Convergence Analysis of Semi-Implicit Euler Methods for Solving Stochastic Age-Dependent Capital System with Variable Delays and Random Jump Magnitudes

We consider semi-implicit Euler methods for stochastic age-dependent capital system with variable delays and random jump magnitudes, and investigate the convergence of the numerical approximation. It is proved that the numerical approximate solutions converge to the analytical solutions in the mean-square sense under given conditions

Markovian Dynamics on Complex Reaction Networks

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm

Properties and advances of probabilistic and statistical algorithms with applications in finance

This thesis is concerned with the construction and enhancement of algorithms involving probability and statistics. The main motivation for these are problems that appear in finance and more generally in applied science. We consider three distinct areas, namely, credit risk modelling, numerics for McKean Vlasov stochastic differential equations and stochastic representations of Partial Differential Equations (PDEs), therefore the thesis is split into three parts. Firstly, we consider the problem of estimating a continuous time Markov chain (CTMC) generator from discrete time observations, which is essentially a missing data problem in statistics. These generators give rise to transition probabilities (in particular probabilities of default) over any time horizon, hence the estimation of such generators is a key problem in the world of banking, where the regulator requires banks to calculate risk over different time horizons. For this particular problem several algorithms have been proposed, however, through a combination of theoretical and numerical results we show the Expectation Maximisation (EM) algorithm to be the superior choice. Furthermore we derive closed form expressions for the associated Wald confidence intervals (error) estimated by the EM algorithm. Previous attempts to calculate such intervals relied on numerical schemes which were slower and less stable. We further provide a closed form expression (via the Delta method) to transfer these errors to the level of the transition probabilities, which are more intuitive. Although one can establish more precise mathematical results with the Markov assumption, there is empirical evidence suggesting this assumption is not valid. We finish this part by carrying out empirical research on non-Markov phenomena and propose a model to capture the so-called rating momentum. This model has many appealing features and is a natural extension to the Markov set up. The second part is based on McKean Vlasov Stochastic Differential Equations (MV-SDEs), these Stochastic Differential Equations (SDEs) arise from looking at the limit, as the number of weakly interacting particles (e.g. gas particles) tends to infinity. The resulting SDE has coefficients which can depend on its own law, making them theoretically more involved. Although MV-SDEs arise from statistical physics, there has been an explosion in interest recently to use MV-SDEs in models for economics. We firstly derive an explicit approximation scheme for MV-SDEs with one-sided Lipschitz growth in the drift. Such a condition was observed to be an issue for standard SDEs and required more sophisticated schemes. There are implicit and explicit schemes one can use and we develop both types in the setting of MV-SDEs. Another main issue for MVSDEs is, due to the dependency on their own law they are extremely expensive to simulate compared to standard SDEs, hence techniques to improve computational cost are in demand. The final result in this part is to develop an importance sampling algorithm for MV-SDEs, where our measure change is obtained through the theory of large deviation principles. Although importance sampling results for standard SDEs are reasonably well understood, there are several difficulties one must overcome to apply a good importance sampling change of measure in this setting. The importance sampling is used here as a variance reduction technique although our results hint that one may be able to use it to reduce propagation of chaos error as well. Finally we consider stochastic algorithms to solve PDEs. It is known one can achieve numerical advantages by using probabilistic methods to solve PDEs, through the so-called probabilistic domain decomposition method. The main result of this part is to present an unbiased stochastic representation for a first order PDE, based on the theory of branching diffusions and regime switching. This is a very interesting result since previously (Itô based) stochastic representations only applied to second order PDEs. There are multiple issues one must overcome in order to obtain an algorithm that is numerically stable and solves such a PDE. We conclude by showing the algorithm’s potential on a more general first order PDE

Hybrid methodology for Markovian epidemic models

In this thesis, we introduce a hybrid discrete-continuous approach suitable for analysing a wide range of epidemiological models, and an approach for improving parameter estimation from data describing the early stages of an outbreak. We restrict our attention to epidemiological models with continuous-time Markov chain (CTMC) dynamics, a ubiquitous framework also commonly used for modelling telecommunication networks, chemical reactions and evolutionary genetics. We introduce our methodology in the framework of the well-known Susceptible–Infectious–Removed (SIR) model, one of the simplest approaches for describing the spread of an infectious disease. We later extend it to a variant of the Susceptible–Exposed–Infectious– Removed (SEIR) model, a generalisation of the SIR CTMC that is more realistic for modelling the initial stage of many outbreaks. Compartmental CTMC models are attractive due to their stochastic individual-to-individual representation of disease transmission. This feature is particularly important when only a small number of infectious individuals are present, during which stage the probability of epidemic fade out is considerable. Unfortunately, the simple SIR CTMC has a state space of order N², where N is the size of the population being modelled, and hence computational limits are quickly reached as N increases. There are a number of approaches towards dealing with this issue, most of which are founded on the principal of restricting one’s attention to the dynamics of the CTMC on a subset of its state space. However, two highly-efficient approaches published in 1970 and 1971 provide a promising alternative to these approaches. The fluid limit [Kurtz, 1970] and diffusion limit [Kurtz, 1971] are large-population approximations of a particular class of CTMC models which approximate the evolution of the underlying CTMC by a deterministic trajectory and a Gaussian diffusion process, respectively. These large-population approximations are governed by a compact system of ordinary differential equations and are suitably accurate so long as the underlying population is sufficiently large. Unfortunately, they become inaccurate if the population of at least one compartment of the underlying CTMC is close to an absorbing boundary, such as during the initial stages of an outbreak. It follows that a natural approach to approximating a CTMC model of a large population is to adopt a hybrid framework, whereby CTMC dynamics are utilised during the initial stages of the outbreak and a suitable large-population approximation is utilised otherwise. In the framework of the SIR CTMC, we present a hybrid fluid model and a hybrid diffusion model which utilise CTMC dynamics while the number of infectious individuals is low and otherwise utilises the fluid limit and the diffusion limit, respectively. We illustrate the utility of our hybrid methodology in computing two key quantities, the distribution of the duration of the outbreak and the distribution of the final size of the outbreak. We demonstrate that the hybrid fluid model provides a suitable approximation of the distribution of the duration of the outbreak and the hybrid diffusion model provides a suitable approximation of the distribution of the final size of the outbreak. In addition, we demonstrate that our hybrid methodology provides a substantial advantage in computational-efficiency over the original SIR CTMC and is superior in accuracy to similar hybrid large-population approaches when considering mid-sized populations. During the initial stages of an outbreak, calibrating a model describing the spread of the disease to the observed data is fundamental to understanding and potentially controlling the disease. A key factor considered by public health officials in planning their response to an outbreak is the transmission potential of the disease, a factor which is informed by estimates of the basic reproductive number, R₀, defined as the average number of secondary cases resulting from a single infectious case in a naive population. However, it is often the case that estimates of R₀ based on data from the initial stages of an outbreak are positively biased. This bias may be the result of various features such as the geography and demography of the outbreak. However, a consideration which is often overlooked is that the outbreak was not detected until such a time as it had established a considerable chain of transmissions, therefore effectively overcoming initial fade out. This is an important feature because the probability of initial fade out is often considerable, making the event that the outbreak becomes established somewhat unlikely. A straightforward way of accounting for this is to condition the model on a particular event, which models the disease overcoming initial fade out. In the framework of both the SIR CTMC and the SEIR CTMC we present a conditioned approach to estimating R₀ from data on the initial stages of an outbreak. For the SIR CTMC, we demonstrate that in certain circumstances, conditioning the model on effectively overcoming initial fade out reduces bias in estimates of R₀ by 0.3 on average, compared to the original CTMC model. Noting that the conditioned model utilises CTMC dynamics throughout, we demonstrate the flexibility of our hybrid methodology by presenting a conditioned hybrid diffusion approach for estimating R₀. We demonstrate that our conditioned hybrid diffusion approach still provides estimates of R₀ which exhibit less bias than under an unconditioned hybrid diffusion model, and that the diffusion methodology enables us to consider larger outbreaks then would have been computationally-feasible in the original conditioned CTMC framework. We demonstrate the flexibility of our conditioned hybrid approach by applying it to a variant of the SEIR CTMC and using it to estimate R₀ from a range of real outbreaks. In so doing, we utilise a truncation rule to ensure the initial CTMC dynamics are computationally-feasible.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 201

METHODS FOR COMPUTATION AND ANALYSIS OF MARKOVIAN DYNAMICS ON COMPLEX NETWORKS

A problem central to many scientific and engineering disciplines is how to deal with noisy dynamic processes that take place on networks. Examples include the ebb and flow of biochemical concentrations within cells, the firing patterns of neurons in the brain, and the spread of disease on social networks. In this thesis, we present a general formalism capable of representing many such problems by means of a master equation. Our study begins by synthesizing the literature to provide a toolkit of known mathematical and computational analysis techniques for dealing with this equation. Subsequently a novel exact numerical solution technique is developed, which can be orders of magnitude faster than the state-of-the-art numerical solver. However, numerical solutions are only applicable to a small subset of processes on networks. Thus, many approximate solution techniques exist in the literature to deal with this problem. Unfortunately, no practical tools exist to quantitatively evaluate the quality of an approximate solution in a given system. Therefore, a statistical tool that is capable of evaluating any analytical or Monte Carlo based approximation to the master equation is developed herein. Finally, we note that larger networks with more complex dynamical phenomena suffer from the same curse of dimensionality as the classical mechanics of a gas. We therefore propose that thermodynamic analysis techniques, adapted from statistical mechanics, may provide a new way forward in analyzing such systems. The investigation focuses on a behavior known as avalanching—complex bursting patterns with fractal properties. By developing thermodynamic analysis techniques along with a potential energy landscape perspective, we are able to demonstrate that increasing intrinsic noise causes a phase transition that results in avalanching. This novel result is utilized to characterize avalanching in an epidemiological model for the first time and to explain avalanching in biological neural networks, in which the cause has been falsely attributed to specific neural architectures. This thesis contributes to the existing literature by providing a novel solution technique, enhances existing and future literature by providing a general method for statistical evaluation of approximative solution techniques, and paves the way towards a promising approach to the thermodynamic analysis of large complex processes on networks