Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

The conditional performance of UK property funds

The evaluation of investment fund performance has been one of the main developments of modern portfolio theory. Most studies employ the technique developed by Jensen (1968) that compares a particular fund's returns to a benchmark portfolio of equal risk. However, the standard measures of fund manager performance are known to suffer from a number of problems in practice. In particular previous studies implicitly assume that the risk level of the portfolio is stationary through the evaluation period. That is unconditional measures of performance do not account for the fact that risk and expected returns may vary with the state of the economy. Therefore many of the problems encountered in previous performance studies reflect the inability of traditional measures to handle the dynamic behaviour of returns. As a consequence Ferson and Schadt (1996) suggest an approach to performance evaluation called conditional performance evaluation which is designed to address this problem. This paper utilises such a conditional measure of performance on a sample of 27 UK property funds, over the period 1987-1998. The results of which suggest that once the time varying nature of the funds beta is corrected for, by the addition of the market indicators, the average fund performance show an improvement over that of the traditional methods of analysis.

Is there a place for property in the multi-asset portfolio?

The case for property has typically rested on the application of modern portfolio theory (MPT), in that property has been shown to offer increased diversification benefits within a multi asset portfolio without hurting portfolio returns especially for lower risk portfolios. However this view is based upon the use of historic, usually appraisal based, data for property. Recent research suggests strongly that such data significantly underestimates the risk characteristics of property, because appraisals explicitly or implicitly smooth out much of the real volatility in property returns. This paper examines the portfolio diversification effects of including property in a multi-asset portfolio, using UK appraisal based (smoothed) data and several derived de-smoothed series. Having considered the effects of de-smoothing, we then consider the inclusion of a further low risk asset (cash) in order to investigate further whether property's place in a low risk portfolio is maintained. The conclusions of this study are that the previous supposed benefits of including property have been overstated. Although property may still have a place in a 'balanced' institutional portfolio, the case for property needs to be reassessed and not be based simplistically on the application of MPT.

The future returns for property required by UK pension funds

In the 1970s Real Estate represented over 17% of the average pension funds total assets. Today such funds hold less than 4%, a figure not seen since the 1960s. This reduction in Real Estate holdings is mainly attributable to the relatively poor performance of Real Estate against other asset classes since the 1980s. Whether pension funds will increase their holding at any point in the future depends therefore on the expected return of Real Estate by comparison with that required to justify a particular asset holding. Using the technique of Modern Portfolio Theory (MPT), this paper assesses the required return that Real Estate would have to offer to justify a 15% holding in a mixed asset portfolio. This figure and the risk/return characteristics of the major asset classes is taken from survey data. Under a number of scenarios it is found that Real Estate can play a part in a mixed asset portfolio at the 15% level. In some cases however, the expected returns of Real Estate are not sufficient to justify a weight of 15% in this asset.

Fenced in: Common property struggles in the management of communal rangelands in central Eastern Cape Province, South Africa

This paper takes as its starting point the assertion that current rangeland management in the central Eastern Cape Province (former Ciskei) of South Africa, is characterised primarily by an ‘open access’ approach. Empirical material drawn from three case-study communities in the region is used to examine the main barriers to management of rangeland as a ‘commons’. The general inability to define and enforce rights to particular grazing resourses in the face of competing claims from ‘outsiders’, as well as inadequate local institutions responsible for rangeland management are highlighted as being of key importance. These are often exacerbated by lack of available grazing land, diffuse user groups and local political and ethnic divisions. Many of these problems have a strong legacy in historical apartheid policies such as forced resettlement and betterment planning. On this basis it is argued that policy should focus on facilitating the emergence of effective, local institutions for rangeland management. Given the limited grazing available to many communities in the region, a critical aspect of this will be finding ways to legitimise current patterns of extensive resource use, which traverse existing ‘community’ boundaries. However, this runs counter to recent legislation, which strongly links community management with legal ownership of land within strict boundaries often defined through fencing. Finding ways to overcome this apparent disjuncture between theory and policy will be vital for the effective management of common pool grazing resources in the region.

Optimal tax theory and the taxation of housing in the US and the UK

First, we survey recent research in the application of optimal tax theory to housing. This work suggests that the under-taxation of housing for owner occupation distorts investment so that owner occupiers are encouraged to over-invest in housing. Simulations of the US economy suggest that this is true there. But, the theoretical work excludes consideration of land and the simulations exclude consideration of taxes other than income taxes. These exclusions are important for the US and UK economies. In the US, the property tax is relatively high. We argue that excluding the property tax is wrong, so that, when the property tax is taken into account, owner occupied housing is not undertaxed in the US. In the UK, property taxes are relatively low but the cost of land has been increasing in real terms for forty years as a result of a policy of constraining land for development. The price of land for housing is now higher than elsewhere. Effectively, an implicit tax is paid by first time buyers which has reduced housing investment. When land is taken into account over-investment in housing is not encouraged in the UK either.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 2. pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy. A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states. The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.