In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on sets measurable in the Caratheodory sense to be finite for a larger class of functions, since the set of all measurable functions with infinite or undefined expected values forms a prevalent subset of the set of all measurable functions, which means "almost all" measurable functions have infinite or undefined expected values. Before we define the specific problem in section 2, we will outline some preliminary definitions. We'll then define the specific problem (along with a partial solution in section 3) to visualize the complete solution. Along the way, we will ask a series of questions to clarify our understanding of the paper.

The effect of capitalization premium in forest estate markets on forest management and climate change mitigation economics is investigated. It is shown that proportional goodwill in capitalization induces linear scaling of the financial return, without any contribution to sound management practices. However, there is a financial discontinuity as harvesting deteriorates goodwill. On the contrary, capitalization premium set on bare land as a tangible asset would increase timber storage and carbon sequestration. Observations indicate that the proportional goodwill is closer to reality within the Nordic Region, resulting in continuity problems but a reduced capital expense for carbon storage.

Two sets of initial conditions are used in the investigation of capital return rate and carbon storage in boreal forests. Firstly, a growth model is applied in young stands as early as the inventory-based model is applicable. Secondly, the growth model is applied to observed wooded stands. Four sets of thinning schedules are investigated in either case. First, the capital return rate is aspired without any restriction. Second, the number of thinnings is restricted to at most one. Third, thinnings are restricted to the removal of only trees thicker than 237 mm. Fourth, commercial thinnings are omitted. The two sets of initial conditions yield similar results. The capital return rate is a weak function of rotation age, which results in variability in the optimal number of thinnings. Reducing the number of thinnings to one increases timber stock but induces a capital return rate deficiency. The deficiency per excess volume unit is smaller if the severity of any thinning is restricted by the removal of large trees only. Omission of thinnings best applies to spruce-dominated stands with stem count less than 2000/ha. Restricted thinning intensity applies to deciduous stands and dense pine stands. The albedo effect increases the benefits of restricted thinnings and increased clearcuttings instead of contradicting the carbon storage.

In this study, a dynamic mixture copula is used to estimate the marginal expected shortfall in the South African insurance sector. While other studies assumed nonlinear dependence to be static over time, our model capture time-varying nonlinear dependence between institutions and the market. In order to capture time-varying nonlinear dependence, the generalized autoregressive score (GAS) is used to model the dynamic copula parameters. Furthermore, our study implements a ranking that expresses to what degree individual insurers are systemically important in South Africa. We use daily stock return of five South African insurers listed in the Johannesburg Stock Exchange (JSE) from November 13, 2007 to June 15, 2020. We find that Sanlam and Discovery contribute the most to systemic risk, while Santam is found to be the least contributor to the overall systemic risk in the South African insurance sector. Our findings would be of paramount importance for the South African regulators as they would be informed that not only banks are systemically important, but some insurers also are systemically important financial institutions. Hence, stricter regulation of these institutions in the form of higher capital and loss absorbency requirements could be required based on the individual business activities undertaken by the company.

Training supervised machine learning models requires labeled examples. A judicious choice of examples is helpful when there is a significant cost associated with assigning labels. This article improves upon a promising extant method – Batch-mode Expected Model Change Maximization (B-EMCM) method – for selecting examples to be labeled for regression problems. Specifically, it develops and evaluates alternate strategies for adaptively selecting batch size in B-EMCM. By determining the cumulative error that occurs from the estimation of the stochastic gradient descent, a stop criteria for each iteration of the batch can be specified to ensure that selected candidates are the most beneficial to model learning. This new methodology is compared to B-EMCM via mean absolute error and root mean square error over ten iterations benchmarked against machine learning data sets. Using multiple data sets and metrics across all methods, one variation of AB-EMCM, the max bound of the accumulated error (AB-EMCM Max), showed the best results for an adaptive batch approach. It achieved better root mean squared error (RMSE) and mean absolute error (MAE) than the other adaptive and non-adaptive batch methods while reaching the result in nearly the same number of iterations as the non-adaptive batch methods.

An empirical model for log yield from trees is established and applied in microeconomics of carbon storage in a boreal spruce estate. The transition from pulpwood to sawlogs is a smoother function of stem diameter in the empirical data, in comparison to literature values. Correspondingly, the value transition of trees along with increasing size is gentler. Due to price premium of sawlogs from clearcuttings, all economically feasible treatment schedules terminate in clearcutting. Best capital return rates are gained with two heavy thinnings from above before clearcutting. Present carbon emission prices allow moderate carbon storage increment if the increment is compensated by proportional carbon rent. Doubling the present carbon prices would allow strong carbon storage increments if compensated by carbon rent. Application of nonproportional carbon rent is proposed.

The expense of carbon sequestration in terms of capital return deficiency is investigated at estate level, in the case of a fertile boreal estate dominated by spruce forest. Thinnings from below result as a high expense of increased rotation age, thinnings from above as a small expense. The expense of increased timber stock is greater than any proportional carbon rent based on present carbon prices. Application of non-proportional carbon rent is proposed.

While the foundations of climate science and ethics are well established, fine-grained climate predictions, as well as policy-decisions, are beset with uncertainties. This chapter maps climate uncertainties and classifies them as to their ground, extent and location. A typology of uncertainty is presented, centered along the axes of scientific and moral uncertainty. This typology is illustrated with paradigmatic examples of uncertainty in climate science, climate ethics and climate economics. Subsequently, the chapter discusses the IPCC’s preferred way of representing uncertainties and evaluates its strengths and weaknesses from a risk management perspective. Three general strategies for decision-makers to cope with climate uncertainty are outlined, the usefulness of which largely depends on whether or not decision-makers find themselves in a context of deep uncertainty. The chapter concludes by offering two recommendations to ease the work of policymakers, faced with the various uncertainties engrained in climate discourse.

Users of meteorological forecasts are often faced with the question of whether to make a decision now based on the current forecast or whether to wait for the next and hopefully more accurate forecast before making the decision. One would imagine that the answer to this question should depend on the extent to which there is a benefit in making the decision now rather than later, combined with an understanding of how the skill of the forecast improves, and information about the possible size and nature of forecast changes. We extend the well-known cost-loss model for forecast-based decision making to capture an idealized version of this situation. We find that within this extended cost-loss model, the question of whether to decide now or wait depends on two specific aspects of the forecast, both of which involve probabilities of probabilities. For the special case of weather and climate forecasts in the form of normal distributions we derive a simulation algorithm, and equivalent analytical expressions, for calculating these two probabilities. We apply the algorithm to forecasts of temperature and find that the algorithm leads to better decisions relative to three simpler alternative decision-making schemes. Similar problems have been studied in many other fields, and we explore some of the connections.

The use of risk measures such as the Value at Risk (VaR) or Tail Conditional Expectation (TCE) is required by the Basel Committee on Banking Supervision in determining a bank’s risk profile. However, both measures can be shown to have shortcomings in the information that they provide to regulators and investors. In this paper we present an introduction to risk measure calculations before demonstrating the weaknesses of these measures. Through the exploration of specific cases we show how familiar yet differing risk profiles have identical values for combinations of these measures. From this evidence we recommend that a sequence of several risk measures should be used to give a more accurate representation of the risk contained on banking balance sheet.

The method propose in this study for stock market multi-objective mathematical model was linear programming using simplex algorithms to optimize the portfolio in the Nigerian Stock Market. The study selected five banks from the list of operators in the market and data were gathered from the banks to have individual market performances over a period of 5years. The data collected contains: Geometric Mean of Monthly Capital Gain Yields and Annual Capital Gain Yields from which Dividend Yields (%) were deducted to arrive Actual rate of return for the banks and the expected rate of return. Furthermore, the risk of semi-absolute deviation below the expected return is reduced. The data was analysed using Python programming because of some clauses in the data gathered. At first, the data assumes integer (-or +) and random in nature. As such, Python programming is one of the software suitable for such solution since the barrier for additivity is broken. Based on the analysis, the study therefore conclude that the potential investor(s) should invest in 13 units of investment x(GTbank Plc), 3 units of First Bank Plc`s Investment (y), 450 units of Zenith bank`s investment (z), and 8 units of Wema bank`s investment (m), and no units of investment of (n) Access bank. These investment quantities will result in the Optimal profit of p= 12797.902 billion naira.

One of the most significant developments in the sports world over the last two decades has been the use of mathematical methods in conjunction with the massive amounts of data now available to analyze performances, identify trends and patterns, and forecast results. Football analytics has advanced significantly in recent years and continues to evolve as it becomes a more recognized and integral part of the game. Football analytics is also used to forecast game outcomes, allowing bettors to make educated guesses. This article describes mathematical concepts related to football analytics that enable a better betting strategies. We explain how the pitch is partitioned into different zones and we define possession sequences. Furthermore, we explain what an expected goals model is and which expected goals model we use in this research. Furthermore, we define two general characteristics of a player evaluation method, each corresponding to one of the equations of the Dawson model. Based on these characteristics, we describe the developments of several general approaches for evaluating players in the context of the Dawson model.

Based on the Expected Utility Theory and Regret Theory, the Extended Regret Theory (ERT) is proposed in this paper to study the optimal pricing strategy of retailers in e-commerce environment. Taking the diversity of sales channels and the uncertainty of consumers in e-commerce environment into consideration, author of the paper designs an extended regret utility function which comprehensively considers both pessimistic and optimistic attitudes of decision makers in retailing industry to describe their regret-avoidance behavior. According to the sensitivity analysis, it is found that the optimal retail price decreases as the consumer price sensitivity coefficient increases, yet does not show variation with changes of the consumers pessimism degree. Moreover, the optimal retail price(s) obtained under EUT, ERT and combination of EUT and ERT represent the same.