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.

Trends in the Australian gender wage differential over the 1980s : some evidence on the effectiveness of legislative reform

The article examines the legislative reforms incorporating the Sex Discrimination Act and the Affirmative Action Act introduced during the 1980s. We utilise the Australian Bureau of Statistics Income Distribution Surveys 1981–82 and 1989–90 to reflect pre- and post-legislative reform. The article adopts the Brown, Moon and Zoloth (1980) methodology which treats both the wage and occupational status of the individual as endogenously determined. In the current context this is a particularly flexible framework allowing one to capture both the direct and indirect effects of the legislative reforms. The indirect effect refers to the narrowing of the gender wage gap associated with legislative manipulation of the male-female occupational distributions. The results contrast the slow convergence in the gender wage gap during the 1980s with the much faster pace of the 1970s. The article concludes that despite the focus of the 1980s legislation on employment equity, changes in the male-female occupational distribution over the period are small and the associated impact on gender wage convergence is also small.

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.

Female concentration and the gender wage differential in the United Kingdom

The paper utilizes the methodology proposed by Johnson and Solon (American Economic Review, 76 (5), 1117-1125, 1986) to examine the impact of job segregation on the gender wage gap in the UK in 1991. The results suggest that despite implementation of the UK 1983 Equal Pay Amendment there remains clear evidence that male/female workers in female dominated jobs continue to earn less for work of ‘similar worth’ than their counterparts in male dominated jobs within the same firm. This conclusion is insensitive to whether one adopts an occupation or firm based measure of gender concentration.

Relative abundance of full-length and truncated FOXP1 isoforms is associated with differential NFκB activity in Follicular Lymphoma

FOXP1 is a transcriptional repressor that has been proposed to repress the expression of some NFκB-responsive genes. Furthermore, truncated forms of FOXP1 have been associated with a subtype of Diffuse Large B-cell Lymphoma characterised by constitutive NFκB activity, indicating that they may inhibit this repression. We have shown that FL tumors have increased relative abundance of truncated FOXP1 isoforms and this is associated with increased expression of NFκB-associated genes. Our results provide strong evidence that relative FOXP1 isoform abundance is associated with NFκB activity in FL, and could potentially be used as a marker for this gene signature.

Differential gene expression in breast cancer cell lines and stroma-tumor differences in microdissected breast cancer biopsies revealed by display array analysis

To examine gene-expression patterning in late-stage breast cancer biopsies, we used a microdissection technique to separate tumor from the surrounding breast tissue or stroma. A DD-PCR protocol was then used to amplify expressed products, which were resolved using PAGE and used as probe to hybridize with representative human arrays and cDNA libraries. The probe derived from the tumor–stroma comparison was hybridized with a gene array and an arrayed cDNA library derived from a GCT of bone; 21 known genes or expressed sequence tags were detected, of which 17 showed differential expression. These included factors associated with epithelial to mesenchymal transition (vimentin), the cargo selection protein (TIP47) and the signal transducer and activator of transcription (STAT3). Northern blot analysis was used to confirm those genes also expressed by representative breast cancer cell lines. Notably, 6 genes of unknown function were restricted to tumor while the majority of stroma-associated genes were known. When applied to transformed breast cancer cell lines (MDA-MB-435 and T47D) that are known to have different metastatic potential, DD array analysis revealed a further 20 genes; 17 of these genes showed differential expression. Use of microdissection and the DD-PCR array protocol allowed us to identify factors whose localized expression within the breast may play a role in abnormal breast development or breast carcinogenesis.