Power Utility Maximization in Discrete-Time and Continuous-Time Exponential Levy Models

Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.Comment: 18 pages, to appear in Mathematical Methods of Operations Research. The final publication is available at springerlink.co

Assessing the role of EO in biodiversity monitoring: options for integrating in-situ observations with EO within the context of the EBONE concept

The European Biodiversity Observation Network (EBONE) is a European contribution on terrestrial monitoring to GEO BON, the Group on Earth Observations Biodiversity Observation Network. EBONE’s aims are to develop a system of biodiversity observation at regional, national and European levels by assessing existing approaches in terms of their validity and applicability starting in Europe, then expanding to regions in Africa. The objective of EBONE is to deliver: 1. A sound scientific basis for the production of statistical estimates of stock and change of key indicators; 2. The development of a system for estimating past changes and forecasting and testing policy options and management strategies for threatened ecosystems and species; 3. A proposal for a cost-effective biodiversity monitoring system. There is a consensus that Earth Observation (EO) has a role to play in monitoring biodiversity. With its capacity to observe detailed spatial patterns and variability across large areas at regular intervals, our instinct suggests that EO could deliver the type of spatial and temporal coverage that is beyond reach with in-situ efforts. Furthermore, when considering the emerging networks of in-situ observations, the prospect of enhancing the quality of the information whilst reducing cost through integration is compelling. This report gives a realistic assessment of the role of EO in biodiversity monitoring and the options for integrating in-situ observations with EO within the context of the EBONE concept (cfr. EBONE-ID1.4). The assessment is mainly based on a set of targeted pilot studies. Building on this assessment, the report then presents a series of recommendations on the best options for using EO in an effective, consistent and sustainable biodiversity monitoring scheme. The issues that we faced were many: 1. Integration can be interpreted in different ways. One possible interpretation is: the combined use of independent data sets to deliver a different but improved data set; another is: the use of one data set to complement another dataset. 2. The targeted improvement will vary with stakeholder group: some will seek for more efficiency, others for more reliable estimates (accuracy and/or precision); others for more detail in space and/or time or more of everything. 3. Integration requires a link between the datasets (EO and in-situ). The strength of the link between reflected electromagnetic radiation and the habitats and their biodiversity observed in-situ is function of many variables, for example: the spatial scale of the observations; timing of the observations; the adopted nomenclature for classification; the complexity of the landscape in terms of composition, spatial structure and the physical environment; the habitat and land cover types under consideration. 4. The type of the EO data available varies (function of e.g. budget, size and location of region, cloudiness, national and/or international investment in airborne campaigns or space technology) which determines its capability to deliver the required output. EO and in-situ could be combined in different ways, depending on the type of integration we wanted to achieve and the targeted improvement. We aimed for an improvement in accuracy (i.e. the reduction in error of our indicator estimate calculated for an environmental zone). Furthermore, EO would also provide the spatial patterns for correlated in-situ data. EBONE in its initial development, focused on three main indicators covering: (i) the extent and change of habitats of European interest in the context of a general habitat assessment; (ii) abundance and distribution of selected species (birds, butterflies and plants); and (iii) fragmentation of natural and semi-natural areas. For habitat extent, we decided that it did not matter how in-situ was integrated with EO as long as we could demonstrate that acceptable accuracies could be achieved and the precision could consistently be improved. The nomenclature used to map habitats in-situ was the General Habitat Classification. We considered the following options where the EO and in-situ play different roles: using in-situ samples to re-calibrate a habitat map independently derived from EO; improving the accuracy of in-situ sampled habitat statistics, by post-stratification with correlated EO data; and using in-situ samples to train the classification of EO data into habitat types where the EO data delivers full coverage or a larger number of samples. For some of the above cases we also considered the impact that the sampling strategy employed to deliver the samples would have on the accuracy and precision achieved. Restricted access to European wide species data prevented work on the indicator ‘abundance and distribution of species’. With respect to the indicator ‘fragmentation’, we investigated ways of delivering EO derived measures of habitat patterns that are meaningful to sampled in-situ observations

Limit cycles in Norwegian lemmings: tensions between phase-dependence and density-dependence

Ever since Elton, the 3–5 year density cycles in lemmings (and other microtines) in Fennoscandia have troubled scientists. Explanations have involved intrinsic regulation and trophic interactions. We have analysed yearly changes in fall abundances for lemmings over 25 years from two local mountain sites in South Norway. These time series appear to have an underlying nonlinear structure of order two. Fitting a piece-wise linear threshold model of maximum order two, the most parsimonious model was, however, of first order for both series. The resulting dynamics from this model is a limit cycle. Reformulating the model in terms of abundances yields a model which combines (delayed) density-dependent effects and the influence of the cyclic phase. The delayed density-dependence of one part of the model is consistent with an effect of specialist predators during the peak and crash phases of the cycle, although other trophic interactions cannot be excluded

Linking climate change to lemming cycles

The population cycles of rodents at northern latitudes have puzzled people for centuries(1,2), and their impact is manifest throughout the alpine ecosystem(2,3). Climate change is known to be able to drive animal population dynamics between stable and cyclic phases(4,5), and has been suggested to cause the recent changes in cyclic dynamics of rodents and their predators(3,6-9). But although predator - rodent interactions are commonly argued to be the cause of the Fennoscandian rodent cycles(1,10-13), the role of the environment in the modulation of such dynamics is often poorly understood in natural systems(8,9,14). Hence, quantitative links between climate-driven processes and rodent dynamics have so far been lacking. Here we show that winter weather and snow conditions, together with density dependence in the net population growth rate, account for the observed population dynamics of the rodent community dominated by lemmings ( Lemmus lemmus) in an alpine Norwegian core habitat between 1970 and 1997, and predict the observed absence of rodent peak years after 1994. These local rodent dynamics are coherent with alpine bird dynamics both locally and over all of southern Norway, consistent with the influence of large- scale fluctuations in winter conditions. The relationship between commonly available meteorological data and snow conditions indicates that changes in temperature and humidity, and thus conditions in the subnivean space, seem to markedly affect the dynamics of alpine rodents and their linked groups. The pattern of less regular rodent peaks, and corresponding changes in the overall dynamics of the alpine ecosystem, thus seems likely to prevail over a growing area under projected climate change

PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS

We consider a financial market with one bond and one stock. The dynamics of the stock price process allow jumps which occur according to a Markov-modulated Poisson process. We assume that there is an investor who is only able to observe the stock price process and not the driving Markov chain. The investor’s aim is to maximize the expected utility of terminal wealth. Using a classical result from filter theory it is possible to reduce this problem with partial observation to one with complete observation. With the help of a generalized Hamilton-Jacobi-Bellman equation where we replace the derivative by Clarke’s generalized gradient, we identify an optimal portfolio strategy. Finally, we discuss some special cases of this model and prove several properties of the optimal portfolio strategy. In particular we derive bounds and discuss the influence of uncertainty on the optimal portfolio strategy. Key words: jump-diffusion process, filtering, utility maximization, stochastic control, generalized HJB equation, optimal portfolio strategies, Bayesian control, stochastic comparison