Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

Numerical solutions of stochastic differential equations – implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

Income Tax : Text, Materials and Essential Cases [8th ed.]

Tax law and policy is a vital part of Australian society. Australian society insists that the Federal Government provide extensive public programs, such as health services, education, social security, foreign aid, legal infra¬structure, regulation, police services, national defence and funding for sports development. These programs are costly to provide and are funded by taxation. The aim of this book is to introduce and explain the principles of tax law and tax policy in plain English. The book contains detailed commentary on tax principles together with extracts from cases and materials that illustrate the application of the principles. The book considers tax policy and the economic and social aspects of tax law. While tax students must develop technical competence in tax law, given the speed with which changes are made to the technical details of tax law, it is also important to grasp tax principles and policy to understand why tax law has changed or why it should change. The chapters are structured to direct readers to the key provisions of the tax law. Each case is introduced by an explanation of the facts, followed by the taxpayer’s arguments, the Commissioner’s assertions and the decision of the Administrative Appeals Tribunal or a court. The commentary guides readers through the issues considered in the judgments. The book contains extracts from: articles; materials dealing with tax policy; and the Commissioner’s rulings. The book also has references for further reading and medium-neutral citations (Internet citations) for cases decided since 1998.

Australia's disabling income support system : tracing the history of the Australian disability income support system 1908 to 2007 : disablism, citizenship and the basic income proposal

The present study examined the historical basis of the Australian disability income support system from 1908 to 2007. Although designed as a safety net for people with a disability, the disability income support system within Australia has been highly targeted. The original eligibility criteria of "permanently incapacitated for work", medical criteria and later "partially capacitated for work" potentially contained ideological inferences that permeated across the time period. This represents an important area for study given the potential consequence for disability income support to marginalise people with a disability. Social policy and disability policy theorists, including Saunders (2007, Social Policy Research Centre [SPRC]) and Gibilisco (2003) have provided valuable insight into some of the effects of disability policy and poverty. Yet while these theorists argued for some form of income support they did not propose a specific form of income security for further exploration. Few studies have undertaken a comprehensive review of the history of disability income support within the Australian context. This thesis sought to redress these gaps by examining disability income support policy within Australia. The research design consisted of an in-depth critical historical-comparative policy analysis methodology. The use of critical historical-comparative policy analysis allowed the researcher to trace the construction of disability within the Australian disability income support policy across four major historical epochs. A framework was developed specifically to guide analysis of the data. The critical discourse analysis method helped to understand the underlying ideological dimensions that led to the predominance of one particular approach over another. Given this, the research purpose of the study centred on: i. Tracing the history of the Australian disability income support system. ii. Examining the historical patterns and ideological assumptions over time. iii. Exploring the historical patterns and ideological assumptions underpinning an alternative model (Basic Income) and the extent to which each model promotes the social citizenship of people with a disability. The research commitment to a social-relational ontology and the quest for social change centred on the idea that "there has to be a better way" in the provision of disability income support. This theme of searching for an alternative reality in disability income support policy resonated throughout the thesis. This thesis found that the Australian disability income support system is disabling in nature and generates categories of disability on the basis of ableness. From the study, ableness became a condition for citizenship. This study acknowledged that, in reality, income support provision reflects only one aspect of the disabling nature of society which requires redressing. Although there are inherent tensions in any redistributive strategy, the Basic Income model potentially provides an alternative to the Australian disability income support system, given its grounding in social citizenship. The thesis findings have implications for academics, policy-makers and practitioners in terms of developing better ways to understand disability constructs in disability income support policy. The thesis also makes a contribution in terms of promoting income support policies based on the rights of all people, not just a few.

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract: Texture enhancement is an important component of image processing, with extensive application in science and engineering. The quality of medical images, quantified using the texture of the images, plays a significant role in the routine diagnosis performed by medical practitioners. Previously, image texture enhancement was performed using classical integral order differential mask operators. Recently, first order fractional differential operators were implemented to enhance images. Experiments conclude that the use of the fractional differential not only maintains the low frequency contour features in the smooth areas of the image, but also nonlinearly enhances edges and textures corresponding to high-frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we applied the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other fractional differential operators, our new algorithms provide higher signal to noise values, which leads to superior image quality.

The N550K/H mutations in FGFR2 confer differential resistance to PD173074, dovitinib and ponatinib ATP-competitive inhibitors

We sought to identify fibroblast growth factor receptor 2 (FGFR2) kinase domain mutations that confer resistance to the pan-FGFR inhibitor, dovitinib, and explore the mechanism of action of the drug-resistant mutations. We cultured BaF3 cells overexpressing FGFR2 in high concentrations of dovitinib and identified fourteen dovitinib-resistant mutations, including the N550K mutation observed in 25% of FGFR2mutant endometrial cancers (EC). Structural and biochemical in vitro kinase analyses, together with BaF3 proliferation assays, showed that the resistance mutations elevate the intrinsic kinase activity of FGFR2. BaF3 lines were used to assess the ability of each mutation to confer cross-resistance to PD173074 and ponatinib. Unlike PD173074, ponatinib effectively inhibited all the dovitinib-resistant FGFR2 mutants except the V565I gatekeeper mutation, suggesting ponatinib but not dovitinib targets the active conformation of FGFR2 kinase. EC cell lines expressing wild-type FGFR2 were relatively resistant to all inhibitors. Whereas EC cell lines expressing mutated FGFR2 showed differential sensitivity. Within the FGFR2mutant cell lines, 3/7 showed marked resistance to PD173074 and relative resistance to dovitinib and ponatinib. This suggests that alternative mechanisms distinct from kinase domain mutations are responsible for intrinsic resistance in these three EC lines. Finally, overexpression of FGFR2N550K in JHUEM-2 cells (FGFR2C383R) conferred resistance (~5 fold) to PD173074, providing independent data that FGFR2N550K can be associated with drug resistance. Biochemical in vitro kinase analyses also shows ponatinib is more effective than dovitinib at inhibiting FGFR2N550K. We propose tumors harboring mutationally activated FGFRs should be treated with FGFR inhibitors that specifically bind the active kinase.

Is quality more important if you’re quirky? A review of the literature on differential susceptibility to childcare environments

Evidence concerning the impact of child care on child development suggests that higher-quality environments, particularly those that are more responsive, predict more favourable social and behavioural outcomes. However, the extent of this effect is not as great as might be expected. Impacts on child outcomes are, at best, modest. One recent explanation emerging from a new theoretical perspective of development, differential susceptibility theory, is that a minority of children are more reactive to both positive and negative environments, while the majority are relatively unaffected. These 'quirky' children have temperamental traits that are more extreme, and are often described in research studies as having 'difficult temperaments'. This paper reviews the literature on such children and argues for the need for further research to identify components of childcare environments that optimise the potential of these more sensitive, quirky individuals.

Double tax agreements and the international allocation of business income in Australia

Increasingly, the effectiveness of the present system of taxation of international businesses is being questioned. The problem associated with the taxation of such businesses is twofold. A system of international taxation must be a fair and equitable system, distributing profits between the relevant jurisdictions and, in doing so, avoiding double taxation. At the same time, the prevention of fiscal evasion must be secured. In an attempt to achieve a fair and equitable system Australia adopts unilateral, bilateral and multilateral measures to avoid double taxation and restrict the avoidance of tax. The first step in ascertaining the international allocation of business income is to consider the taxation of business income according to domestic law, that is, the unilateral measures. The treatment of international business income under the Australian domestic law, that is, the Income Tax Assessment Act 1936 (Cth) and Income Tax Assessment Act 1997 (Cth), will depend on two concepts, first, whether the taxpayer is a resident of Australia and secondly, whether the income is sourced in Australia. After the taxation of business profits has been determined according to domestic law it is necessary to consider the applicability of the bilateral measures, that is, the Double Tax Agreements (DTAs) to which Australia is a party, as the DTAs will override the domestic law where there is any conflict. Australia is a party to 40 DTAs with another seven presently being negotiated. The preamble to Australia's DTAs provides that the purpose of such agreements is 'to conclude an Agreement for the avoidance of double taxation and the prevention of fiscal evasion with respect to taxes on income'. Both purposes, for different reasons, are equally important. It has been said that: The taxpayer hopes the treaty will prevent the double taxation of his income; the tax gatherer hopes the treaty will prevent fiscal evasion; and the politician just hopes. The first purpose, the avoidance of double taxation, is achieved through the provision of rules whereby the Contracting States agree to the classification of income and the allocation of that income to a particular State. In this sense DTAs do not allocate jurisdiction to tax but rather provide an arrangement whereby the States agree to restrict their substantive law. The restriction is either through the non-taxing of the income or via the provision of a tax credit.

Australia - taxation of trusts – the problem of aligning concepts of income

The Australian government is currently considering options for the rewrite and reform of the current provisions which apply to the taxation of trust income. This article provides a discussion of the current regime and the proposed reforms. It is suggested that a major revamp of taxation of trust income in Australia is problematic and a simpler approach may be to leave the law as is, with modification where necessary to address key issues as and when they arise.