Future Drawdown Research Articles

On occupation times in the red of Lévy risk models

In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level 0) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only the Laplace transforms are known. Due to the close connection between occupation time and many other quantities, we provide a few applications of our results including future drawdown, inverse occupation time, Parisian ruin with exponential delay, and the last time at running maximum.

On future drawdowns of Lévy processes

For a given Lévy process X=(Xt)t∈R+ and for fixed s∈R+∪{∞} and t∈R+ we analyse the future drawdown extremes that are defined as follows: D¯t,s∗=sup0≤u≤tinfu≤w<t+s(Xw−Xu),D¯t,s∗=inf0≤u≤tinfu≤w<t+s(Xw−Xu). The path-functionals D¯t,s∗ and D¯t,s∗ are of interest in various areas of application, including financial mathematics and queueing theory. In the case that X has a strictly positive mean, we find the exact asymptotic decay as x→∞ of the tail probabilities P(D¯t∗<x) and P(D¯t∗<x) of D¯t∗=lims→∞D¯t,s∗ and D¯t∗=lims→∞D¯t,s∗ both when the jumps satisfy the Cramér assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the Lévy process X are of single sign and X is not subordinator, we identify the one-dimensional distributions in terms of the scale function of X. By way of example, we derive explicit results for the Black–Scholes–Samuelson model.

Water Resources Evaluation in Ghadamis Basin, Libya

Ghadames basin is one of the important groundwater basins in Libya. This basin covers an area of about 215,000 km2 in northwestern Libya. This groundwater basin consists of two major aquifer systems. The upper aquifer system mostly formed of carbonate rocks (Mizdah and Nalut formations). The lower aquifer system formed mostly of sandstone known as Kiklah formation. The two aquifer systems are separated by semi-confining layers known as Yefren marl. The lower aquifer system is considered as the main aquifer for water supply. The water resource (mostly from the lower aquifer system) in these systems is used for domestic, irrigation and fodder drinking. The domestic use of water is to supply water to the cities and towns in western part of the country and in the northern flanks of the basin, and plans to transport water to the coastal cities in northwestern part of the country. The existing total amount of extracted water is 132 Mm3/year, and about 20% of it used for domestic uses. The planned water extraction is 223.8 Mm3/year about 55% of it will be used for domestic uses. The hydrogeological situation for this basin has been studied in this study using data from previous hydrogeological exploration programs, evaluation of the data from the existing extraction and observation water wells, pumping tests data used to evaluate aquifers hydraulic parameters, and construction of hydrogeological cross-sections to know the aquifer systems geometry. Hydogeological computer model was constructed to evaluate the aquifer hydraulic parameters using the historical water extraction, and water levels and drawdowns. The model is then used to predict the future drawdown which has been found to be in the range 55 to 60 m after 50 years of extraction.

Cape Around the World: Update 2014 The Relationship between Risk and Return

We update our annual analysis of expected returns for 38 equity markets around the world. Based on current CAPE valuations, we expect mid to high single-digit yearly returns in most developed equity markets. However, risks are elevated for some markets. We specifically investigate drawdown risk and find that given its current valuation levels, the US market in particular exhibits a high risk of significant future drawdowns.

Prediction of future drawdown of water levels of the Pleistocene aquifer system of Wadi El-Assiuti Area, Eastern Desert, Egypt, as a criterion for management and conservation

Wadi El-Assiuti designates a remarkable dry drainage basin in Middle Egypt. The Pleistocene groundwater aquifer of this area represents the main water source used for reclamation and domestic purposes. The increasing number of groundwater wells drilled at the entrance of Wadi El-Assiuti in recent years, with its consequent over consumption of the groundwater reservoir, seems to be the clear-cut cause for groundwater levels drawdown and quantity deterioration. The constructed flow net map for the Pleistocene aquifer system sketched a pattern of piezometric pressures and flow directions. The regional water flow is from east–northeast towards west–southwest directions, with some adverse minor directions at the north-central part of the study area. This map declared the areas suffering from over pumping (piezometric lows). However, using empirical formulae and assumptions, the expected drop of piezometric levels of groundwater in the Pleistocene aquifer, as future scenarios, was predicted for 5, 10 and 20 years. These future scenarios reflected the urgent need for a policy or management scheme to decrease the severe drawdown of water level in this important developmental area.

The great manmade river project: A partial solution to Libya's future water supply

The southern part of Libya is a desert area of negligible population density overlying two of the largest ground‐water basins in the world: the Murzuk basin in the south‐west and Kufra‐Sarir basin in the south‐east. In situ utilization of these huge groundwater reserves is uneconomical due to poor soils and unsuitable climatic conditions. Extensive hydrogeological studies conducted during the last two decades have indicated the possibility of transferring over 6 million m3 per day to the coastal zones in the north. The Great Manmade River Project was launched in 1983 aiming at a rational utilization of the transported water for agricultural and urban developments, along with restoration of the affected aquifers. The project consists of five phases and is planned to be completed before the turn of the century. This article highlights the technical aspects of this major project in terms of water availability, water quality, future drawdowns, cost of water, well‐field design and other hydrogeological paramet...

Nye metoder for prøvepumpning af boringer og grundvandsreservoirer

The purpose of this paper is to introduce the practical application of selected quantitative and analytical methods for testing wells and evaluation of aquifers to engineers and geologists within the field of groundwater hydrology in the Nordic countries. This abstract, a list of symbols and units and text of figures and tables in English may possibly provide understanding of the used formulas and methods for readers not familiar with Danish.The increasing demand for ground water and the need for optimal utilization of aquifers accelerate the use of quantitative methods for predictions of available quantities of ground water and aquifer response on withdrawal.The first part of this paper gives a short introduction into the developed theory for analysing drawdown data from pumping tests and a number of selected formulas in their developed form for determining the hydraulic properties of wells (B, C and n), aquifers, (T and S) and confining bed (P'). Selected hydrogeological terms are defined. The procedure of pumping and the frequency of waterlevel observations in space and time is briefly mentioned both for step-drawdown test and pumping test with constant capacity.Formulas for adjustments of drawdown- and recovery data for barometric efficiency and decrease of saturated thickness are described. Examples of logarithmic and semilogarithmic plots of data are given together with type curves for leaky artesian and watertable conditions.Principles and formulas for prediction of the future drawdown in pumped wells and aquifers are outlined.The second part of the paper gives the practical application of the selected analytical methods and formulas. Illustrative examples with actual field data collected by the Hydrogeological Department, Geological Survey of Denmark, are presented for each method discussed. Basic data as well drawdown as recovery from 4 pumping tests and their numerical and graphical analysis a represented in 27 figures and 16 tables.

Safe Yield of a Well Field in a Leaky‐Artesian Strip Aquifera

ABSTRACTCoarse granular deposits in preglacial river valleys are an important source of ground water on the Canadian prairies. Such an aquifer can commonly be modeled by an infinite‐strip leaky‐artesian aquifer. Test results can be analyzed, future drawdowns predicted, and safe yields estimated by applying the standard leaky‐artesian formula in conjunction with image‐well theory, making due allowance for well‐loss factors.This paper develops basic formulas required for safe yield estimation for an array of n wells located on the axis of an infinite‐strip leaky‐artesian aquifer. In general, determination of safe yield for each well depends on the solution of n simultaneous linear or nonlinear equations. For certain symmetrical arrays, however, the number of equations to be solved is approximately halved. A linear equation will apply for any well if flow adjacent to the well bore always obeys the Darcy law for all pumping rates of interest. If all n equations are linear, they may be solved by the methods of matrix algebra; if not, a trial‐and‐error solution must be adopted.The method is illustrated by an application to a 6‐mile stretch of a buried‐valley aquifer near Edson, Alberta. The numerical coefficients in the system of equations are first derived and the equations then solved to obtain estimated 20‐year safe yields for well arrays containing up to 38 equally spaced wells.

Mutual Interference of Water Wells

A general equation is developed which describes the relationship between the drawdown of any one well in an array which is pumping at a decreasing rate, the duration of pumping, the distance from the well to the other wells, and the aquifer characteristics for infinite, homogeneous, confined aquifer. The decrease in discharge is introduced by means of the head-capacity curve for each pump. These relationships can be used to predict future drawdown and discharge for any individual well, existing or proposed, in an array having a given geometrical pattern. An example is provided for the case of three wells located at the vertices of an equilateral triangle. The variation of discharge with time for various well spacings for each of the three wells is presented.