Nonparametric estimation for Levy processes with a view towards mathematical finance

Nonparametric methods for the estimation of the Levy density of a Levy process are developed. Estimators that can be written in terms of the ``jumps'' of the process are introduced, and so are discrete-data based approximations. A model selection approach made up of two steps is investigated. The first step consists in the selection of a good estimator from a linear model of proposed Levy densities, while the second is a data-driven selection of a linear model among a given collection of linear models. By providing lower bounds for the minimax risk of estimation over Besov Levy densities, our estimators are shown to achieve the ``best'' rate of convergence. A numerical study for the case of histogram estimators and for variance Gamma processes, models of key importance in risky asset price modeling driven by Levy processes, is presented.Comment: 68 pages, 19 figures, submitted to Annals of Statistic

The Standard Model Higgs as the origin of the hot Big Bang

If the Standard Model (SM) Higgs is weakly coupled to the inflationary sector, the Higgs is expected to be universally in the form of a condensate towards the end of inflation. The Higgs decays rapidly after inflation - via non-perturbative effects - into an out-of-equilibrium distribution of SM species, which thermalize soon afterwards. If the post-inflationary equation of state of the universe is stiff, $w \simeq +1$, the SM species eventually dominate the total energy budget. This provides a natural origin for the relativistic thermal plasma of SM species, required for the onset of the `hot Big Bang' era. The viability of this scenario requires the inflationary Hubble scale $H_*$ to be lower than the instability scale for Higgs vacuum decay, the Higgs not to generate too large curvature perturbations at cosmological scales, and the SM dominance to occur before Big Bang Nucleosynthesis. We show that successful reheating into the SM can only be obtained in the presence of a non-minimal coupling to gravity $\xi \gtrsim 1$, with a reheating temperature of $T_{\rm RH} \gtrsim \mathcal{O}(10^{10})\xi^{3/2}(H_*/10^{14}{\rm GeV})^2~{\rm GeV}$.Comment: 6 pages, 2 figures, minor changes with new figures to match published version in PL

High-order short-time expansions for ATM option prices of exponential L\'evy models

In the present work, a novel second-order approximation for ATM option prices is derived for a large class of exponential L\'{e}vy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second-order term, in time-$t$, is of the form $d_{2}\,t^{(3-Y)/2}$, with $d_{2}$ only depending on $Y$, the degree of jump activity, on $\sigma$, the volatility of the continuous component, and on an additional parameter controlling the intensity of the "small" jumps (regardless of their signs). This extends the well known result that the leading first-order term is $\sigma t^{1/2}/\sqrt{2\pi}$. In contrast, under a pure-jump model, the dependence on $Y$ and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form $d_{1}t^{1/Y}$. The second-order term is shown to be of the form $\tilde{d}_{2} t$ and, therefore, its order of decay turns out to be independent of $Y$. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. Our approach is sufficiently general to cover a wide class of L\'{e}vy processes which satisfy the latter property and whose L\'{e}vy densitiy can be closely approximated by a stable density near the origin. Our numerical results show that the first-order term typically exhibits rather poor performance and that the second-order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.Comment: 35 pages, 8 figures. This is an extension of our earlier submission arXiv:1112.3111. To appear in Mathematical Financ

Influence of mycorrhizal fungi on soil geochemistry and plant uptake of cadmium and zinc: greenhouse and soil studies of Palmerton Zinc Superfund Site, Pennsylvania, u.s.a

The application of bioremediation as a means to ameliorate potentially toxic metals from contaminated soils has been on the forefront of sustainable technologies in the field of environmental studies. Conventional methods available for soil remediation are employed in both in situ or ex situ and these are: soil flushing, solidification/stabilization, vitrification, electrokinetics, chemical reduction/oxidation, soil washing, and excavation. In general, these technologies are limited by their applicability to small areas and the high costs to execute. The use of biota, in particular fungi and trees, has been an emerging low-cost and environmentally-friendly alternative to conventional technologies. In this study, I evaluate four soils collected from the Lehigh Gap Wildlife Refuge and Nature Center adjacent to the Palmerton Zinc Superfund Site by using commonly used single extraction methods and the revised European Bureau of Common References (BCR) sequential extraction to determine availability of Cd and Zn. I also explore how the rhizosphere-based method (RBM) that mimics rhizosphere conditions compares to the liquid cultures of the ectomycorrhizal fungus Paxillus involutus to interpret change in metal availability and the potential for environmental risk associated with fungal inoculation of soils when used to aid remediation projects. Additionally, comparison between extractants (EDTA, DTPA, Mehlich-III, Fraction 1 and Pseudototal of the revised BCR sequential extraction method) that evaluate plant availability to determine the efficacy of the previous mentioned extractants. Findings were that both components of the BCR method grossly overestimates both Cd and Zn compared to aforementioned extractants. Additionally, fungal liquid performed similarly to the Mehlich-III as comparable extractant for determining Cd and Zn availability, where fungal liquid and RBM had similar extractability for Zn only.;Work was also done to investigate effects of P. involutus and a Populus hybrid on fractionation of metals in these soils by the revised BCR sequential extraction method. Findings suggest that availability and mobility of Cd and Zn in contaminated soils can be altered significantly by P. involutus and Populus and plant uptake of Cd is greater than Zn. With further research with more varieties of hybrid poplars and mycorrhizal fungi more effective symbionts can be chosen to aid in remediating metal contaminated sites that may pose a risk to the environment and human health